tight piecewise convex relaxations for global optimization of … · – exploit the narrow bounds...

Post on 23-Jul-2020

1 Views

Category:

Documents

0 Downloads

Preview:

Click to see full reader

TRANSCRIPT

Tight Piecewise Convex Relaxations for Global Optimization of Optimal Power Flow

Harsha NagarajanLos Alamos National Laboratory

Mowen Lu & Russell BentDiscussions with Prof. J. Linderoth

Jan 10, 2019

LA-UR-19-20884

Optimal Power Flow• Introduced in 1962

• The basis for many of the economic decisions made by modern grid operators

• Rising interest in solving AC OPF• DOE ARPA-E Go Competition

https://gocompetition.energy.gov

• Generator dispatch, Unit commitment, Transmission switching, etc.

• AC OPF is NP hard (Bienstock, Verma - 2006)

Optimal Power Flow

Minimize cost of generation

Kirchhoff ’s law

Ohm’s law

Engineering limits

Convex relaxations for OPF

SDP relaxations– Bai, Wei, Fujisawa, Wang (2008)– R. Madani, S. Sojoudi, J. Lavaei (2014) SOCQC

SDPAC

Sgi ´ Sd

i “ÿ

pi,jqPEYER

Sij @i P N

Sij “ Y ˚ij ViV

˚i ´ Y ˚

ij ViV˚j pi, jq P E Y ER

SOC-based relaxations– R. Jabr (2006) – B. Kocuk, S. S. Dey, and X. A. Sun (2016)

Convex quadratic (QC) relaxations– H. Hijazi, C. Coffrin, and P. Van Hentenryck (2015)

Source: C. Coffrin, et. al. “The QC relaxation: A theoretical and computational study on optimal power flow”, 2016

And many others

QC-relaxation overviewWii = v2i i 2 N

<(Wij) = vivj cos(✓i � ✓j) 8(i, j) 2 E

=(Wij) = vivj sin(✓i � ✓j) 8(i, j) 2 E

(1)

‣ Key ideas– factorable-functions relaxation– exploit the narrow bounds in power systems– convexify transcendental functions (sin, cos)

‣ Resulting optimization model– quadratic and convex (computationally better)

H. Hijazi, C. Coffrin, P.V. Hentenryck, Convex Quadratic Relaxations for Mixed-Integer Nonlinear Programs in Power Systems, Math. Prog.-C, 2015

QC-relaxation overviewWii = v2i i 2 N

<(Wij) = vivj cos(✓i � ✓j) 8(i, j) 2 E

=(Wij) = vivj sin(✓i � ✓j) 8(i, j) 2 E

(1)

Recursive McCormick relaxation

Trilinear monomials

Wii = hv2i iT i 2 N

<(Wij) = hhvivjiM hcos(✓i � ✓j)iCiM 8(i, j) 2 E

=(Wij) = hhvivjiM hsin(✓i � ✓j)iSiM 8(i, j) 2 E

QC-relaxation overviewWii = v2i i 2 N

<(Wij) = vivj cos(✓i � ✓j) 8(i, j) 2 E

=(Wij) = vivj sin(✓i � ✓j) 8(i, j) 2 E

(1)

Trilinear monomials

H-representation

�10

1 �1 �0.5 0 0.5 1

�1

0

1

vivj

dv ivj

Convex Hull of Bilinear Function

<latexit sha1_base64="mQmST6YwzgY4OoW1dSuHhNKq3JA=">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</latexit><latexit sha1_base64="mQmST6YwzgY4OoW1dSuHhNKq3JA=">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</latexit><latexit sha1_base64="mQmST6YwzgY4OoW1dSuHhNKq3JA=">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</latexit><latexit sha1_base64="mQmST6YwzgY4OoW1dSuHhNKq3JA=">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</latexit>

dvivj > vlivj + vljvi � vlivlj

dvivj > vui vj + vuj vi � vui vuj

dvivj 6 vlivj + vuj vi � vlivuj

dvivj 6 vui vj + vljvi � vui vlj

<latexit sha1_base64="hO8CBTxZEWuriKv6/Bs7b1j0T24=">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</latexit><latexit sha1_base64="hO8CBTxZEWuriKv6/Bs7b1j0T24=">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</latexit><latexit sha1_base64="hO8CBTxZEWuriKv6/Bs7b1j0T24=">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</latexit><latexit sha1_base64="hO8CBTxZEWuriKv6/Bs7b1j0T24=">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</latexit>

Recursive McCormick relaxation

QC-relaxation overviewWii = v2i i 2 N

<(Wij) = vivj cos(✓i � ✓j) 8(i, j) 2 E

=(Wij) = vivj sin(✓i � ✓j) 8(i, j) 2 E

(1)

Trilinear monomials

H-representation

Apply recursively on

May not capture it’s convex hull

(asymmetric bounds on voltage and phase-

angle variables)

dvivj > vlivj + vljvi � vlivlj

dvivj > vui vj + vuj vi � vui vuj

dvivj 6 vlivj + vuj vi � vlivuj

dvivj 6 vui vj + vljvi � vui vlj

<latexit sha1_base64="hO8CBTxZEWuriKv6/Bs7b1j0T24=">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</latexit><latexit sha1_base64="hO8CBTxZEWuriKv6/Bs7b1j0T24=">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</latexit><latexit sha1_base64="hO8CBTxZEWuriKv6/Bs7b1j0T24=">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</latexit><latexit sha1_base64="hO8CBTxZEWuriKv6/Bs7b1j0T24=">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</latexit>

Recursive McCormick relaxation

Recursive vs. Convex hull relaxationsSymmetric bounds:

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7hx1x2x3x4i - gap

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

hhx

1x

2ix

3ix

4-

gap

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7hx1x2x3x4i - gap

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

hx1x

2ih

x3x

4i-

gap

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7hx1x2x3x4i - gap

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x1hx

2hx

3x

4ii

-ga

p

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7hx1x2x3x4i - gap

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

hx1hx

2x

3ii

x4

-ga

p

hhx1x2ix3ix4<latexit sha1_base64="qFmyxzyQ0NanKf/EVexK2WRqTF0=">AAACFXicbZBNSwMxEIaz9avWr1WPXoJF8CBltxb0WPTisYL9gHZZsulsG5rNLklWLKV/wot/xYsHRbwK3vw3pu2K2jow5Mk7MyTzBglnSjvOp5VbWl5ZXcuvFzY2t7Z37N29hopTSaFOYx7LVkAUcCagrpnm0EokkCjg0AwGl5N68xakYrG40cMEvIj0BAsZJdpIvn3S4UT0OGQHvvNdk+WO/L6e/mDFt4tOyZkGXgQ3gyLKoubbH51uTNMIhKacKNV2nUR7IyI1oxzGhU6qICF0QHrQNihIBMobTbca4yOjdHEYS5NC46n6e2JEIqWGUWA6I6L7ar42Ef+rtVMdnnsjJpJUg6Czh8KUYx3jiUW4yyRQzYcGCJXM/BXTPpGEamNkwZjgzq+8CI1yyTV8XSlWLzI78ugAHaJj5KIzVEVXqIbqiKJ79Iie0Yv1YD1Zr9bbrDVnZTP76E9Y719qDZ5R</latexit><latexit sha1_base64="qFmyxzyQ0NanKf/EVexK2WRqTF0=">AAACFXicbZBNSwMxEIaz9avWr1WPXoJF8CBltxb0WPTisYL9gHZZsulsG5rNLklWLKV/wot/xYsHRbwK3vw3pu2K2jow5Mk7MyTzBglnSjvOp5VbWl5ZXcuvFzY2t7Z37N29hopTSaFOYx7LVkAUcCagrpnm0EokkCjg0AwGl5N68xakYrG40cMEvIj0BAsZJdpIvn3S4UT0OGQHvvNdk+WO/L6e/mDFt4tOyZkGXgQ3gyLKoubbH51uTNMIhKacKNV2nUR7IyI1oxzGhU6qICF0QHrQNihIBMobTbca4yOjdHEYS5NC46n6e2JEIqWGUWA6I6L7ar42Ef+rtVMdnnsjJpJUg6Czh8KUYx3jiUW4yyRQzYcGCJXM/BXTPpGEamNkwZjgzq+8CI1yyTV8XSlWLzI78ugAHaJj5KIzVEVXqIbqiKJ79Iie0Yv1YD1Zr9bbrDVnZTP76E9Y719qDZ5R</latexit><latexit sha1_base64="qFmyxzyQ0NanKf/EVexK2WRqTF0=">AAACFXicbZBNSwMxEIaz9avWr1WPXoJF8CBltxb0WPTisYL9gHZZsulsG5rNLklWLKV/wot/xYsHRbwK3vw3pu2K2jow5Mk7MyTzBglnSjvOp5VbWl5ZXcuvFzY2t7Z37N29hopTSaFOYx7LVkAUcCagrpnm0EokkCjg0AwGl5N68xakYrG40cMEvIj0BAsZJdpIvn3S4UT0OGQHvvNdk+WO/L6e/mDFt4tOyZkGXgQ3gyLKoubbH51uTNMIhKacKNV2nUR7IyI1oxzGhU6qICF0QHrQNihIBMobTbca4yOjdHEYS5NC46n6e2JEIqWGUWA6I6L7ar42Ef+rtVMdnnsjJpJUg6Czh8KUYx3jiUW4yyRQzYcGCJXM/BXTPpGEamNkwZjgzq+8CI1yyTV8XSlWLzI78ugAHaJj5KIzVEVXqIbqiKJ79Iie0Yv1YD1Zr9bbrDVnZTP76E9Y719qDZ5R</latexit><latexit sha1_base64="qFmyxzyQ0NanKf/EVexK2WRqTF0=">AAACFXicbZBNSwMxEIaz9avWr1WPXoJF8CBltxb0WPTisYL9gHZZsulsG5rNLklWLKV/wot/xYsHRbwK3vw3pu2K2jow5Mk7MyTzBglnSjvOp5VbWl5ZXcuvFzY2t7Z37N29hopTSaFOYx7LVkAUcCagrpnm0EokkCjg0AwGl5N68xakYrG40cMEvIj0BAsZJdpIvn3S4UT0OGQHvvNdk+WO/L6e/mDFt4tOyZkGXgQ3gyLKoubbH51uTNMIhKacKNV2nUR7IyI1oxzGhU6qICF0QHrQNihIBMobTbca4yOjdHEYS5NC46n6e2JEIqWGUWA6I6L7ar42Ef+rtVMdnnsjJpJUg6Czh8KUYx3jiUW4yyRQzYcGCJXM/BXTPpGEamNkwZjgzq+8CI1yyTV8XSlWLzI78ugAHaJj5KIzVEVXqIbqiKJ79Iie0Yv1YD1Zr9bbrDVnZTP76E9Y719qDZ5R</latexit>

hx1x2ihx3x4i<latexit sha1_base64="5UMGEnPYKyovYrDRNVwaksyAD60=">AAACFnicbVBNS8NAEJ3Ur1q/oh69LBbBiyWpBT0WvXisYFuhDWGz3bRLN5uwuxFL6K/w4l/x4kERr+LNf+O2DaitDwbevjfD7Lwg4Uxpx/myCkvLK6trxfXSxubW9o69u9dScSoJbZKYx/I2wIpyJmhTM83pbSIpjgJO28HwcuK376hULBY3epRQL8J9wUJGsDaSb590ORZ9TtG975qqduXs+SOfmqrlsm+XnYozBVokbk7KkKPh25/dXkzSiApNOFaq4zqJ9jIsNSOcjkvdVNEEkyHu046hAkdUedn0rDE6MkoPhbE0JTSaqr8nMhwpNYoC0xlhPVDz3kT8z+ukOjz3MiaSVFNBZovClCMdo0lGqMckJZqPDMFEMvNXRAZYYqJNkiUTgjt/8iJpVSuu4de1cv0ij6MIB3AIx+DCGdThChrQBAIP8AQv8Go9Ws/Wm/U+ay1Y+cw+/IH18Q3BXZ57</latexit><latexit sha1_base64="5UMGEnPYKyovYrDRNVwaksyAD60=">AAACFnicbVBNS8NAEJ3Ur1q/oh69LBbBiyWpBT0WvXisYFuhDWGz3bRLN5uwuxFL6K/w4l/x4kERr+LNf+O2DaitDwbevjfD7Lwg4Uxpx/myCkvLK6trxfXSxubW9o69u9dScSoJbZKYx/I2wIpyJmhTM83pbSIpjgJO28HwcuK376hULBY3epRQL8J9wUJGsDaSb590ORZ9TtG975qqduXs+SOfmqrlsm+XnYozBVokbk7KkKPh25/dXkzSiApNOFaq4zqJ9jIsNSOcjkvdVNEEkyHu046hAkdUedn0rDE6MkoPhbE0JTSaqr8nMhwpNYoC0xlhPVDz3kT8z+ukOjz3MiaSVFNBZovClCMdo0lGqMckJZqPDMFEMvNXRAZYYqJNkiUTgjt/8iJpVSuu4de1cv0ij6MIB3AIx+DCGdThChrQBAIP8AQv8Go9Ws/Wm/U+ay1Y+cw+/IH18Q3BXZ57</latexit><latexit sha1_base64="5UMGEnPYKyovYrDRNVwaksyAD60=">AAACFnicbVBNS8NAEJ3Ur1q/oh69LBbBiyWpBT0WvXisYFuhDWGz3bRLN5uwuxFL6K/w4l/x4kERr+LNf+O2DaitDwbevjfD7Lwg4Uxpx/myCkvLK6trxfXSxubW9o69u9dScSoJbZKYx/I2wIpyJmhTM83pbSIpjgJO28HwcuK376hULBY3epRQL8J9wUJGsDaSb590ORZ9TtG975qqduXs+SOfmqrlsm+XnYozBVokbk7KkKPh25/dXkzSiApNOFaq4zqJ9jIsNSOcjkvdVNEEkyHu046hAkdUedn0rDE6MkoPhbE0JTSaqr8nMhwpNYoC0xlhPVDz3kT8z+ukOjz3MiaSVFNBZovClCMdo0lGqMckJZqPDMFEMvNXRAZYYqJNkiUTgjt/8iJpVSuu4de1cv0ij6MIB3AIx+DCGdThChrQBAIP8AQv8Go9Ws/Wm/U+ay1Y+cw+/IH18Q3BXZ57</latexit><latexit sha1_base64="5UMGEnPYKyovYrDRNVwaksyAD60=">AAACFnicbVBNS8NAEJ3Ur1q/oh69LBbBiyWpBT0WvXisYFuhDWGz3bRLN5uwuxFL6K/w4l/x4kERr+LNf+O2DaitDwbevjfD7Lwg4Uxpx/myCkvLK6trxfXSxubW9o69u9dScSoJbZKYx/I2wIpyJmhTM83pbSIpjgJO28HwcuK376hULBY3epRQL8J9wUJGsDaSb590ORZ9TtG975qqduXs+SOfmqrlsm+XnYozBVokbk7KkKPh25/dXkzSiApNOFaq4zqJ9jIsNSOcjkvdVNEEkyHu046hAkdUedn0rDE6MkoPhbE0JTSaqr8nMhwpNYoC0xlhPVDz3kT8z+ukOjz3MiaSVFNBZovClCMdo0lGqMckJZqPDMFEMvNXRAZYYqJNkiUTgjt/8iJpVSuu4de1cv0ij6MIB3AIx+DCGdThChrQBAIP8AQv8Go9Ws/Wm/U+ay1Y+cw+/IH18Q3BXZ57</latexit>

x1hx2hx3x4ii<latexit sha1_base64="Uuxciw8UsOTqwiKW0JPmWWCD2UA=">AAACFnicbZBNSwMxEIZn61etX6sevQSL4MWyWwt6LHrxWMG2Qrss2TTbhmazS5IVy9Jf4cW/4sWDIl7Fm//GtF1QWweGPHlnhmTeIOFMacf5sgpLyyura8X10sbm1vaOvbvXUnEqCW2SmMfyNsCKciZoUzPN6W0iKY4CTtvB8HJSb99RqVgsbvQooV6E+4KFjGBtJN8+Qfe+2+VY9Dk1WP3BU5O1rpxd89O3y07FmQZaBDeHMuTR8O3Pbi8maUSFJhwr1XGdRHsZlpoRTselbqpogskQ92nHoMARVV42XWuMjozSQ2EsTQqNpurviQxHSo2iwHRGWA/UfG0i/lfrpDo89zImklRTQWYPhSlHOkYTj1CPSUo0HxnARDLzV0QGWGKijZMlY4I7v/IitKoV1/B1rVy/yO0owgEcwjG4cAZ1uIIGNIHAAzzBC7xaj9az9Wa9z1oLVj6zD3/C+vgGuv6eew==</latexit><latexit sha1_base64="Uuxciw8UsOTqwiKW0JPmWWCD2UA=">AAACFnicbZBNSwMxEIZn61etX6sevQSL4MWyWwt6LHrxWMG2Qrss2TTbhmazS5IVy9Jf4cW/4sWDIl7Fm//GtF1QWweGPHlnhmTeIOFMacf5sgpLyyura8X10sbm1vaOvbvXUnEqCW2SmMfyNsCKciZoUzPN6W0iKY4CTtvB8HJSb99RqVgsbvQooV6E+4KFjGBtJN8+Qfe+2+VY9Dk1WP3BU5O1rpxd89O3y07FmQZaBDeHMuTR8O3Pbi8maUSFJhwr1XGdRHsZlpoRTselbqpogskQ92nHoMARVV42XWuMjozSQ2EsTQqNpurviQxHSo2iwHRGWA/UfG0i/lfrpDo89zImklRTQWYPhSlHOkYTj1CPSUo0HxnARDLzV0QGWGKijZMlY4I7v/IitKoV1/B1rVy/yO0owgEcwjG4cAZ1uIIGNIHAAzzBC7xaj9az9Wa9z1oLVj6zD3/C+vgGuv6eew==</latexit><latexit sha1_base64="Uuxciw8UsOTqwiKW0JPmWWCD2UA=">AAACFnicbZBNSwMxEIZn61etX6sevQSL4MWyWwt6LHrxWMG2Qrss2TTbhmazS5IVy9Jf4cW/4sWDIl7Fm//GtF1QWweGPHlnhmTeIOFMacf5sgpLyyura8X10sbm1vaOvbvXUnEqCW2SmMfyNsCKciZoUzPN6W0iKY4CTtvB8HJSb99RqVgsbvQooV6E+4KFjGBtJN8+Qfe+2+VY9Dk1WP3BU5O1rpxd89O3y07FmQZaBDeHMuTR8O3Pbi8maUSFJhwr1XGdRHsZlpoRTselbqpogskQ92nHoMARVV42XWuMjozSQ2EsTQqNpurviQxHSo2iwHRGWA/UfG0i/lfrpDo89zImklRTQWYPhSlHOkYTj1CPSUo0HxnARDLzV0QGWGKijZMlY4I7v/IitKoV1/B1rVy/yO0owgEcwjG4cAZ1uIIGNIHAAzzBC7xaj9az9Wa9z1oLVj6zD3/C+vgGuv6eew==</latexit><latexit sha1_base64="Uuxciw8UsOTqwiKW0JPmWWCD2UA=">AAACFnicbZBNSwMxEIZn61etX6sevQSL4MWyWwt6LHrxWMG2Qrss2TTbhmazS5IVy9Jf4cW/4sWDIl7Fm//GtF1QWweGPHlnhmTeIOFMacf5sgpLyyura8X10sbm1vaOvbvXUnEqCW2SmMfyNsCKciZoUzPN6W0iKY4CTtvB8HJSb99RqVgsbvQooV6E+4KFjGBtJN8+Qfe+2+VY9Dk1WP3BU5O1rpxd89O3y07FmQZaBDeHMuTR8O3Pbi8maUSFJhwr1XGdRHsZlpoRTselbqpogskQ92nHoMARVV42XWuMjozSQ2EsTQqNpurviQxHSo2iwHRGWA/UfG0i/lfrpDo89zImklRTQWYPhSlHOkYTj1CPSUo0HxnARDLzV0QGWGKijZMlY4I7v/IitKoV1/B1rVy/yO0owgEcwjG4cAZ1uIIGNIHAAzzBC7xaj9az9Wa9z1oLVj6zD3/C+vgGuv6eew==</latexit>

hx1hx2x3iix4<latexit sha1_base64="GV0nnRieEiA8w6Kt1HFqSIjTB3c=">AAACF3icbZBLSwMxEMez9VXrq+rRS7AInspuLeix6MVjBfuA7rJk09k2NJtdkqy0LP0WXvwqXjwo4lVvfhvTh6itA0N++c8MyfyDhDOlbfvTyq2srq1v5DcLW9s7u3vF/YOmilNJoUFjHst2QBRwJqChmebQTiSQKODQCgZXk3rrDqRisbjVowS8iPQECxkl2kh+sexyInoc8NB3frBi8gy7cnb/Pod+1S+W7LI9DbwMzhxKaB51v/jhdmOaRiA05USpjmMn2suI1IxyGBfcVEFC6ID0oGNQkAiUl033GuMTo3RxGEuTQuOp+nsiI5FSoygwnRHRfbVYm4j/1TqpDi+8jIkk1SDo7KEw5VjHeGIS7jIJVPORAUIlM3/FtE8kodpYWTAmOIsrL0OzUnYM31RLtcu5HXl0hI7RKXLQOaqha1RHDUTRPXpEz+jFerCerFfrbdaas+Yzh+hPWO9fKEWepQ==</latexit><latexit sha1_base64="GV0nnRieEiA8w6Kt1HFqSIjTB3c=">AAACF3icbZBLSwMxEMez9VXrq+rRS7AInspuLeix6MVjBfuA7rJk09k2NJtdkqy0LP0WXvwqXjwo4lVvfhvTh6itA0N++c8MyfyDhDOlbfvTyq2srq1v5DcLW9s7u3vF/YOmilNJoUFjHst2QBRwJqChmebQTiSQKODQCgZXk3rrDqRisbjVowS8iPQECxkl2kh+sexyInoc8NB3frBi8gy7cnb/Pod+1S+W7LI9DbwMzhxKaB51v/jhdmOaRiA05USpjmMn2suI1IxyGBfcVEFC6ID0oGNQkAiUl033GuMTo3RxGEuTQuOp+nsiI5FSoygwnRHRfbVYm4j/1TqpDi+8jIkk1SDo7KEw5VjHeGIS7jIJVPORAUIlM3/FtE8kodpYWTAmOIsrL0OzUnYM31RLtcu5HXl0hI7RKXLQOaqha1RHDUTRPXpEz+jFerCerFfrbdaas+Yzh+hPWO9fKEWepQ==</latexit><latexit sha1_base64="GV0nnRieEiA8w6Kt1HFqSIjTB3c=">AAACF3icbZBLSwMxEMez9VXrq+rRS7AInspuLeix6MVjBfuA7rJk09k2NJtdkqy0LP0WXvwqXjwo4lVvfhvTh6itA0N++c8MyfyDhDOlbfvTyq2srq1v5DcLW9s7u3vF/YOmilNJoUFjHst2QBRwJqChmebQTiSQKODQCgZXk3rrDqRisbjVowS8iPQECxkl2kh+sexyInoc8NB3frBi8gy7cnb/Pod+1S+W7LI9DbwMzhxKaB51v/jhdmOaRiA05USpjmMn2suI1IxyGBfcVEFC6ID0oGNQkAiUl033GuMTo3RxGEuTQuOp+nsiI5FSoygwnRHRfbVYm4j/1TqpDi+8jIkk1SDo7KEw5VjHeGIS7jIJVPORAUIlM3/FtE8kodpYWTAmOIsrL0OzUnYM31RLtcu5HXl0hI7RKXLQOaqha1RHDUTRPXpEz+jFerCerFfrbdaas+Yzh+hPWO9fKEWepQ==</latexit><latexit sha1_base64="GV0nnRieEiA8w6Kt1HFqSIjTB3c=">AAACF3icbZBLSwMxEMez9VXrq+rRS7AInspuLeix6MVjBfuA7rJk09k2NJtdkqy0LP0WXvwqXjwo4lVvfhvTh6itA0N++c8MyfyDhDOlbfvTyq2srq1v5DcLW9s7u3vF/YOmilNJoUFjHst2QBRwJqChmebQTiSQKODQCgZXk3rrDqRisbjVowS8iPQECxkl2kh+sexyInoc8NB3frBi8gy7cnb/Pod+1S+W7LI9DbwMzhxKaB51v/jhdmOaRiA05USpjmMn2suI1IxyGBfcVEFC6ID0oGNQkAiUl033GuMTo3RxGEuTQuOp+nsiI5FSoygwnRHRfbVYm4j/1TqpDi+8jIkk1SDo7KEw5VjHeGIS7jIJVPORAUIlM3/FtE8kodpYWTAmOIsrL0OzUnYM31RLtcu5HXl0hI7RKXLQOaqha1RHDUTRPXpEz+jFerCerFfrbdaas+Yzh+hPWO9fKEWepQ==</latexit>

xi 2 [�U,U ]<latexit sha1_base64="DYw7mq4IFUNT1he+U9YaIKQdBiU=">AAAB9XicbZDNSsNAFIVv6l+tf1GXbgaL4EJLIoIui25cVjBtIY1lMp20QyeTMDNRS+h7uHGhiFvfxZ1v47TNQlsPDHycey/3zglTzpR2nG+rtLS8srpWXq9sbG5t79i7e02VZJJQjyQ8ke0QK8qZoJ5mmtN2KimOQ05b4fB6Um89UKlYIu70KKVBjPuCRYxgbaz7py5DHSaQf+qdeEHXrjo1Zyq0CG4BVSjU6NpfnV5CspgKTThWynedVAc5lpoRTseVTqZoiskQ96lvUOCYqiCfXj1GR8bpoSiR5gmNpu7viRzHSo3i0HTGWA/UfG1i/lfzMx1dBjkTaaapILNFUcaRTtAkAtRjkhLNRwYwkczcisgAS0y0CapiQnDnv7wIzbOaa/j2vFq/KuIowwEcwjG4cAF1uIEGeEBAwjO8wpv1aL1Y79bHrLVkFTP78EfW5w/md5F0</latexit><latexit sha1_base64="DYw7mq4IFUNT1he+U9YaIKQdBiU=">AAAB9XicbZDNSsNAFIVv6l+tf1GXbgaL4EJLIoIui25cVjBtIY1lMp20QyeTMDNRS+h7uHGhiFvfxZ1v47TNQlsPDHycey/3zglTzpR2nG+rtLS8srpWXq9sbG5t79i7e02VZJJQjyQ8ke0QK8qZoJ5mmtN2KimOQ05b4fB6Um89UKlYIu70KKVBjPuCRYxgbaz7py5DHSaQf+qdeEHXrjo1Zyq0CG4BVSjU6NpfnV5CspgKTThWynedVAc5lpoRTseVTqZoiskQ96lvUOCYqiCfXj1GR8bpoSiR5gmNpu7viRzHSo3i0HTGWA/UfG1i/lfzMx1dBjkTaaapILNFUcaRTtAkAtRjkhLNRwYwkczcisgAS0y0CapiQnDnv7wIzbOaa/j2vFq/KuIowwEcwjG4cAF1uIEGeEBAwjO8wpv1aL1Y79bHrLVkFTP78EfW5w/md5F0</latexit><latexit sha1_base64="DYw7mq4IFUNT1he+U9YaIKQdBiU=">AAAB9XicbZDNSsNAFIVv6l+tf1GXbgaL4EJLIoIui25cVjBtIY1lMp20QyeTMDNRS+h7uHGhiFvfxZ1v47TNQlsPDHycey/3zglTzpR2nG+rtLS8srpWXq9sbG5t79i7e02VZJJQjyQ8ke0QK8qZoJ5mmtN2KimOQ05b4fB6Um89UKlYIu70KKVBjPuCRYxgbaz7py5DHSaQf+qdeEHXrjo1Zyq0CG4BVSjU6NpfnV5CspgKTThWynedVAc5lpoRTseVTqZoiskQ96lvUOCYqiCfXj1GR8bpoSiR5gmNpu7viRzHSo3i0HTGWA/UfG1i/lfzMx1dBjkTaaapILNFUcaRTtAkAtRjkhLNRwYwkczcisgAS0y0CapiQnDnv7wIzbOaa/j2vFq/KuIowwEcwjG4cAF1uIEGeEBAwjO8wpv1aL1Y79bHrLVkFTP78EfW5w/md5F0</latexit><latexit sha1_base64="DYw7mq4IFUNT1he+U9YaIKQdBiU=">AAAB9XicbZDNSsNAFIVv6l+tf1GXbgaL4EJLIoIui25cVjBtIY1lMp20QyeTMDNRS+h7uHGhiFvfxZ1v47TNQlsPDHycey/3zglTzpR2nG+rtLS8srpWXq9sbG5t79i7e02VZJJQjyQ8ke0QK8qZoJ5mmtN2KimOQ05b4fB6Um89UKlYIu70KKVBjPuCRYxgbaz7py5DHSaQf+qdeEHXrjo1Zyq0CG4BVSjU6NpfnV5CspgKTThWynedVAc5lpoRTseVTqZoiskQ96lvUOCYqiCfXj1GR8bpoSiR5gmNpu7viRzHSo3i0HTGWA/UfG1i/lfzMx1dBjkTaaapILNFUcaRTtAkAtRjkhLNRwYwkczcisgAS0y0CapiQnDnv7wIzbOaa/j2vFq/KuIowwEcwjG4cAF1uIEGeEBAwjO8wpv1aL1Y79bHrLVkFTP78EfW5w/md5F0</latexit>

Recursive vs. Convex hull relaxationsAsymmetric bounds:

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014hx1x2x3x4i - gap

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

hhx

1x

2ix

3ix

4-

gap

0.000 0.002 0.004 0.006 0.008 0.010hx1x2x3x4i - gap

0.0

0.1

0.2

0.3

0.4

0.5

hx1x

2ih

x3x

4i-

gap

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014hx1x2x3x4i - gap

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

x1hx

2hx

3x

4ii

-ga

p

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014hx1x2x3x4i - gap

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08hx

1hx

2x

3ii

x4

-ga

p

hhx1x2ix3ix4<latexit sha1_base64="qFmyxzyQ0NanKf/EVexK2WRqTF0=">AAACFXicbZBNSwMxEIaz9avWr1WPXoJF8CBltxb0WPTisYL9gHZZsulsG5rNLklWLKV/wot/xYsHRbwK3vw3pu2K2jow5Mk7MyTzBglnSjvOp5VbWl5ZXcuvFzY2t7Z37N29hopTSaFOYx7LVkAUcCagrpnm0EokkCjg0AwGl5N68xakYrG40cMEvIj0BAsZJdpIvn3S4UT0OGQHvvNdk+WO/L6e/mDFt4tOyZkGXgQ3gyLKoubbH51uTNMIhKacKNV2nUR7IyI1oxzGhU6qICF0QHrQNihIBMobTbca4yOjdHEYS5NC46n6e2JEIqWGUWA6I6L7ar42Ef+rtVMdnnsjJpJUg6Czh8KUYx3jiUW4yyRQzYcGCJXM/BXTPpGEamNkwZjgzq+8CI1yyTV8XSlWLzI78ugAHaJj5KIzVEVXqIbqiKJ79Iie0Yv1YD1Zr9bbrDVnZTP76E9Y719qDZ5R</latexit><latexit sha1_base64="qFmyxzyQ0NanKf/EVexK2WRqTF0=">AAACFXicbZBNSwMxEIaz9avWr1WPXoJF8CBltxb0WPTisYL9gHZZsulsG5rNLklWLKV/wot/xYsHRbwK3vw3pu2K2jow5Mk7MyTzBglnSjvOp5VbWl5ZXcuvFzY2t7Z37N29hopTSaFOYx7LVkAUcCagrpnm0EokkCjg0AwGl5N68xakYrG40cMEvIj0BAsZJdpIvn3S4UT0OGQHvvNdk+WO/L6e/mDFt4tOyZkGXgQ3gyLKoubbH51uTNMIhKacKNV2nUR7IyI1oxzGhU6qICF0QHrQNihIBMobTbca4yOjdHEYS5NC46n6e2JEIqWGUWA6I6L7ar42Ef+rtVMdnnsjJpJUg6Czh8KUYx3jiUW4yyRQzYcGCJXM/BXTPpGEamNkwZjgzq+8CI1yyTV8XSlWLzI78ugAHaJj5KIzVEVXqIbqiKJ79Iie0Yv1YD1Zr9bbrDVnZTP76E9Y719qDZ5R</latexit><latexit sha1_base64="qFmyxzyQ0NanKf/EVexK2WRqTF0=">AAACFXicbZBNSwMxEIaz9avWr1WPXoJF8CBltxb0WPTisYL9gHZZsulsG5rNLklWLKV/wot/xYsHRbwK3vw3pu2K2jow5Mk7MyTzBglnSjvOp5VbWl5ZXcuvFzY2t7Z37N29hopTSaFOYx7LVkAUcCagrpnm0EokkCjg0AwGl5N68xakYrG40cMEvIj0BAsZJdpIvn3S4UT0OGQHvvNdk+WO/L6e/mDFt4tOyZkGXgQ3gyLKoubbH51uTNMIhKacKNV2nUR7IyI1oxzGhU6qICF0QHrQNihIBMobTbca4yOjdHEYS5NC46n6e2JEIqWGUWA6I6L7ar42Ef+rtVMdnnsjJpJUg6Czh8KUYx3jiUW4yyRQzYcGCJXM/BXTPpGEamNkwZjgzq+8CI1yyTV8XSlWLzI78ugAHaJj5KIzVEVXqIbqiKJ79Iie0Yv1YD1Zr9bbrDVnZTP76E9Y719qDZ5R</latexit><latexit sha1_base64="qFmyxzyQ0NanKf/EVexK2WRqTF0=">AAACFXicbZBNSwMxEIaz9avWr1WPXoJF8CBltxb0WPTisYL9gHZZsulsG5rNLklWLKV/wot/xYsHRbwK3vw3pu2K2jow5Mk7MyTzBglnSjvOp5VbWl5ZXcuvFzY2t7Z37N29hopTSaFOYx7LVkAUcCagrpnm0EokkCjg0AwGl5N68xakYrG40cMEvIj0BAsZJdpIvn3S4UT0OGQHvvNdk+WO/L6e/mDFt4tOyZkGXgQ3gyLKoubbH51uTNMIhKacKNV2nUR7IyI1oxzGhU6qICF0QHrQNihIBMobTbca4yOjdHEYS5NC46n6e2JEIqWGUWA6I6L7ar42Ef+rtVMdnnsjJpJUg6Czh8KUYx3jiUW4yyRQzYcGCJXM/BXTPpGEamNkwZjgzq+8CI1yyTV8XSlWLzI78ugAHaJj5KIzVEVXqIbqiKJ79Iie0Yv1YD1Zr9bbrDVnZTP76E9Y719qDZ5R</latexit>

hx1x2ihx3x4i<latexit sha1_base64="5UMGEnPYKyovYrDRNVwaksyAD60=">AAACFnicbVBNS8NAEJ3Ur1q/oh69LBbBiyWpBT0WvXisYFuhDWGz3bRLN5uwuxFL6K/w4l/x4kERr+LNf+O2DaitDwbevjfD7Lwg4Uxpx/myCkvLK6trxfXSxubW9o69u9dScSoJbZKYx/I2wIpyJmhTM83pbSIpjgJO28HwcuK376hULBY3epRQL8J9wUJGsDaSb590ORZ9TtG975qqduXs+SOfmqrlsm+XnYozBVokbk7KkKPh25/dXkzSiApNOFaq4zqJ9jIsNSOcjkvdVNEEkyHu046hAkdUedn0rDE6MkoPhbE0JTSaqr8nMhwpNYoC0xlhPVDz3kT8z+ukOjz3MiaSVFNBZovClCMdo0lGqMckJZqPDMFEMvNXRAZYYqJNkiUTgjt/8iJpVSuu4de1cv0ij6MIB3AIx+DCGdThChrQBAIP8AQv8Go9Ws/Wm/U+ay1Y+cw+/IH18Q3BXZ57</latexit><latexit sha1_base64="5UMGEnPYKyovYrDRNVwaksyAD60=">AAACFnicbVBNS8NAEJ3Ur1q/oh69LBbBiyWpBT0WvXisYFuhDWGz3bRLN5uwuxFL6K/w4l/x4kERr+LNf+O2DaitDwbevjfD7Lwg4Uxpx/myCkvLK6trxfXSxubW9o69u9dScSoJbZKYx/I2wIpyJmhTM83pbSIpjgJO28HwcuK376hULBY3epRQL8J9wUJGsDaSb590ORZ9TtG975qqduXs+SOfmqrlsm+XnYozBVokbk7KkKPh25/dXkzSiApNOFaq4zqJ9jIsNSOcjkvdVNEEkyHu046hAkdUedn0rDE6MkoPhbE0JTSaqr8nMhwpNYoC0xlhPVDz3kT8z+ukOjz3MiaSVFNBZovClCMdo0lGqMckJZqPDMFEMvNXRAZYYqJNkiUTgjt/8iJpVSuu4de1cv0ij6MIB3AIx+DCGdThChrQBAIP8AQv8Go9Ws/Wm/U+ay1Y+cw+/IH18Q3BXZ57</latexit><latexit sha1_base64="5UMGEnPYKyovYrDRNVwaksyAD60=">AAACFnicbVBNS8NAEJ3Ur1q/oh69LBbBiyWpBT0WvXisYFuhDWGz3bRLN5uwuxFL6K/w4l/x4kERr+LNf+O2DaitDwbevjfD7Lwg4Uxpx/myCkvLK6trxfXSxubW9o69u9dScSoJbZKYx/I2wIpyJmhTM83pbSIpjgJO28HwcuK376hULBY3epRQL8J9wUJGsDaSb590ORZ9TtG975qqduXs+SOfmqrlsm+XnYozBVokbk7KkKPh25/dXkzSiApNOFaq4zqJ9jIsNSOcjkvdVNEEkyHu046hAkdUedn0rDE6MkoPhbE0JTSaqr8nMhwpNYoC0xlhPVDz3kT8z+ukOjz3MiaSVFNBZovClCMdo0lGqMckJZqPDMFEMvNXRAZYYqJNkiUTgjt/8iJpVSuu4de1cv0ij6MIB3AIx+DCGdThChrQBAIP8AQv8Go9Ws/Wm/U+ay1Y+cw+/IH18Q3BXZ57</latexit><latexit sha1_base64="5UMGEnPYKyovYrDRNVwaksyAD60=">AAACFnicbVBNS8NAEJ3Ur1q/oh69LBbBiyWpBT0WvXisYFuhDWGz3bRLN5uwuxFL6K/w4l/x4kERr+LNf+O2DaitDwbevjfD7Lwg4Uxpx/myCkvLK6trxfXSxubW9o69u9dScSoJbZKYx/I2wIpyJmhTM83pbSIpjgJO28HwcuK376hULBY3epRQL8J9wUJGsDaSb590ORZ9TtG975qqduXs+SOfmqrlsm+XnYozBVokbk7KkKPh25/dXkzSiApNOFaq4zqJ9jIsNSOcjkvdVNEEkyHu046hAkdUedn0rDE6MkoPhbE0JTSaqr8nMhwpNYoC0xlhPVDz3kT8z+ukOjz3MiaSVFNBZovClCMdo0lGqMckJZqPDMFEMvNXRAZYYqJNkiUTgjt/8iJpVSuu4de1cv0ij6MIB3AIx+DCGdThChrQBAIP8AQv8Go9Ws/Wm/U+ay1Y+cw+/IH18Q3BXZ57</latexit>

x1hx2hx3x4ii<latexit sha1_base64="Uuxciw8UsOTqwiKW0JPmWWCD2UA=">AAACFnicbZBNSwMxEIZn61etX6sevQSL4MWyWwt6LHrxWMG2Qrss2TTbhmazS5IVy9Jf4cW/4sWDIl7Fm//GtF1QWweGPHlnhmTeIOFMacf5sgpLyyura8X10sbm1vaOvbvXUnEqCW2SmMfyNsCKciZoUzPN6W0iKY4CTtvB8HJSb99RqVgsbvQooV6E+4KFjGBtJN8+Qfe+2+VY9Dk1WP3BU5O1rpxd89O3y07FmQZaBDeHMuTR8O3Pbi8maUSFJhwr1XGdRHsZlpoRTselbqpogskQ92nHoMARVV42XWuMjozSQ2EsTQqNpurviQxHSo2iwHRGWA/UfG0i/lfrpDo89zImklRTQWYPhSlHOkYTj1CPSUo0HxnARDLzV0QGWGKijZMlY4I7v/IitKoV1/B1rVy/yO0owgEcwjG4cAZ1uIIGNIHAAzzBC7xaj9az9Wa9z1oLVj6zD3/C+vgGuv6eew==</latexit><latexit sha1_base64="Uuxciw8UsOTqwiKW0JPmWWCD2UA=">AAACFnicbZBNSwMxEIZn61etX6sevQSL4MWyWwt6LHrxWMG2Qrss2TTbhmazS5IVy9Jf4cW/4sWDIl7Fm//GtF1QWweGPHlnhmTeIOFMacf5sgpLyyura8X10sbm1vaOvbvXUnEqCW2SmMfyNsCKciZoUzPN6W0iKY4CTtvB8HJSb99RqVgsbvQooV6E+4KFjGBtJN8+Qfe+2+VY9Dk1WP3BU5O1rpxd89O3y07FmQZaBDeHMuTR8O3Pbi8maUSFJhwr1XGdRHsZlpoRTselbqpogskQ92nHoMARVV42XWuMjozSQ2EsTQqNpurviQxHSo2iwHRGWA/UfG0i/lfrpDo89zImklRTQWYPhSlHOkYTj1CPSUo0HxnARDLzV0QGWGKijZMlY4I7v/IitKoV1/B1rVy/yO0owgEcwjG4cAZ1uIIGNIHAAzzBC7xaj9az9Wa9z1oLVj6zD3/C+vgGuv6eew==</latexit><latexit sha1_base64="Uuxciw8UsOTqwiKW0JPmWWCD2UA=">AAACFnicbZBNSwMxEIZn61etX6sevQSL4MWyWwt6LHrxWMG2Qrss2TTbhmazS5IVy9Jf4cW/4sWDIl7Fm//GtF1QWweGPHlnhmTeIOFMacf5sgpLyyura8X10sbm1vaOvbvXUnEqCW2SmMfyNsCKciZoUzPN6W0iKY4CTtvB8HJSb99RqVgsbvQooV6E+4KFjGBtJN8+Qfe+2+VY9Dk1WP3BU5O1rpxd89O3y07FmQZaBDeHMuTR8O3Pbi8maUSFJhwr1XGdRHsZlpoRTselbqpogskQ92nHoMARVV42XWuMjozSQ2EsTQqNpurviQxHSo2iwHRGWA/UfG0i/lfrpDo89zImklRTQWYPhSlHOkYTj1CPSUo0HxnARDLzV0QGWGKijZMlY4I7v/IitKoV1/B1rVy/yO0owgEcwjG4cAZ1uIIGNIHAAzzBC7xaj9az9Wa9z1oLVj6zD3/C+vgGuv6eew==</latexit><latexit sha1_base64="Uuxciw8UsOTqwiKW0JPmWWCD2UA=">AAACFnicbZBNSwMxEIZn61etX6sevQSL4MWyWwt6LHrxWMG2Qrss2TTbhmazS5IVy9Jf4cW/4sWDIl7Fm//GtF1QWweGPHlnhmTeIOFMacf5sgpLyyura8X10sbm1vaOvbvXUnEqCW2SmMfyNsCKciZoUzPN6W0iKY4CTtvB8HJSb99RqVgsbvQooV6E+4KFjGBtJN8+Qfe+2+VY9Dk1WP3BU5O1rpxd89O3y07FmQZaBDeHMuTR8O3Pbi8maUSFJhwr1XGdRHsZlpoRTselbqpogskQ92nHoMARVV42XWuMjozSQ2EsTQqNpurviQxHSo2iwHRGWA/UfG0i/lfrpDo89zImklRTQWYPhSlHOkYTj1CPSUo0HxnARDLzV0QGWGKijZMlY4I7v/IitKoV1/B1rVy/yO0owgEcwjG4cAZ1uIIGNIHAAzzBC7xaj9az9Wa9z1oLVj6zD3/C+vgGuv6eew==</latexit>

hx1hx2x3iix4<latexit sha1_base64="GV0nnRieEiA8w6Kt1HFqSIjTB3c=">AAACF3icbZBLSwMxEMez9VXrq+rRS7AInspuLeix6MVjBfuA7rJk09k2NJtdkqy0LP0WXvwqXjwo4lVvfhvTh6itA0N++c8MyfyDhDOlbfvTyq2srq1v5DcLW9s7u3vF/YOmilNJoUFjHst2QBRwJqChmebQTiSQKODQCgZXk3rrDqRisbjVowS8iPQECxkl2kh+sexyInoc8NB3frBi8gy7cnb/Pod+1S+W7LI9DbwMzhxKaB51v/jhdmOaRiA05USpjmMn2suI1IxyGBfcVEFC6ID0oGNQkAiUl033GuMTo3RxGEuTQuOp+nsiI5FSoygwnRHRfbVYm4j/1TqpDi+8jIkk1SDo7KEw5VjHeGIS7jIJVPORAUIlM3/FtE8kodpYWTAmOIsrL0OzUnYM31RLtcu5HXl0hI7RKXLQOaqha1RHDUTRPXpEz+jFerCerFfrbdaas+Yzh+hPWO9fKEWepQ==</latexit><latexit sha1_base64="GV0nnRieEiA8w6Kt1HFqSIjTB3c=">AAACF3icbZBLSwMxEMez9VXrq+rRS7AInspuLeix6MVjBfuA7rJk09k2NJtdkqy0LP0WXvwqXjwo4lVvfhvTh6itA0N++c8MyfyDhDOlbfvTyq2srq1v5DcLW9s7u3vF/YOmilNJoUFjHst2QBRwJqChmebQTiSQKODQCgZXk3rrDqRisbjVowS8iPQECxkl2kh+sexyInoc8NB3frBi8gy7cnb/Pod+1S+W7LI9DbwMzhxKaB51v/jhdmOaRiA05USpjmMn2suI1IxyGBfcVEFC6ID0oGNQkAiUl033GuMTo3RxGEuTQuOp+nsiI5FSoygwnRHRfbVYm4j/1TqpDi+8jIkk1SDo7KEw5VjHeGIS7jIJVPORAUIlM3/FtE8kodpYWTAmOIsrL0OzUnYM31RLtcu5HXl0hI7RKXLQOaqha1RHDUTRPXpEz+jFerCerFfrbdaas+Yzh+hPWO9fKEWepQ==</latexit><latexit sha1_base64="GV0nnRieEiA8w6Kt1HFqSIjTB3c=">AAACF3icbZBLSwMxEMez9VXrq+rRS7AInspuLeix6MVjBfuA7rJk09k2NJtdkqy0LP0WXvwqXjwo4lVvfhvTh6itA0N++c8MyfyDhDOlbfvTyq2srq1v5DcLW9s7u3vF/YOmilNJoUFjHst2QBRwJqChmebQTiSQKODQCgZXk3rrDqRisbjVowS8iPQECxkl2kh+sexyInoc8NB3frBi8gy7cnb/Pod+1S+W7LI9DbwMzhxKaB51v/jhdmOaRiA05USpjmMn2suI1IxyGBfcVEFC6ID0oGNQkAiUl033GuMTo3RxGEuTQuOp+nsiI5FSoygwnRHRfbVYm4j/1TqpDi+8jIkk1SDo7KEw5VjHeGIS7jIJVPORAUIlM3/FtE8kodpYWTAmOIsrL0OzUnYM31RLtcu5HXl0hI7RKXLQOaqha1RHDUTRPXpEz+jFerCerFfrbdaas+Yzh+hPWO9fKEWepQ==</latexit><latexit sha1_base64="GV0nnRieEiA8w6Kt1HFqSIjTB3c=">AAACF3icbZBLSwMxEMez9VXrq+rRS7AInspuLeix6MVjBfuA7rJk09k2NJtdkqy0LP0WXvwqXjwo4lVvfhvTh6itA0N++c8MyfyDhDOlbfvTyq2srq1v5DcLW9s7u3vF/YOmilNJoUFjHst2QBRwJqChmebQTiSQKODQCgZXk3rrDqRisbjVowS8iPQECxkl2kh+sexyInoc8NB3frBi8gy7cnb/Pod+1S+W7LI9DbwMzhxKaB51v/jhdmOaRiA05USpjmMn2suI1IxyGBfcVEFC6ID0oGNQkAiUl033GuMTo3RxGEuTQuOp+nsiI5FSoygwnRHRfbVYm4j/1TqpDi+8jIkk1SDo7KEw5VjHeGIS7jIJVPORAUIlM3/FtE8kodpYWTAmOIsrL0OzUnYM31RLtcu5HXl0hI7RKXLQOaqha1RHDUTRPXpEz+jFerCerFfrbdaas+Yzh+hPWO9fKEWepQ==</latexit>

xi 2 [�U1, U2]<latexit sha1_base64="HjV70tkeR6sIrNLGcwZd9iqouBI=">AAAB+3icbZDNSsNAFIVv/K31L9alm8EiuNCSFEGXRTcuK5i2kIYwmU7aoZNJmJlIS+mruHGhiFtfxJ1v47TNQlsPDHycey/3zokyzpR2nG9rbX1jc2u7tFPe3ds/OLSPKi2V5pJQj6Q8lZ0IK8qZoJ5mmtNOJilOIk7b0fBuVm8/UalYKh71OKNBgvuCxYxgbazQroxChrpMIP/SC90LL6wHoV11as5caBXcAqpQqBnaX91eSvKECk04Vsp3nUwHEyw1I5xOy91c0QyTIe5T36DACVXBZH77FJ0Zp4fiVJonNJq7vycmOFFqnESmM8F6oJZrM/O/mp/r+CaYMJHlmgqyWBTnHOkUzYJAPSYp0XxsABPJzK2IDLDERJu4yiYEd/nLq9Cq11zDD1fVxm0RRwlO4BTOwYVraMA9NMEDAiN4hld4s6bWi/VufSxa16xi5hj+yPr8Abwfku4=</latexit><latexit sha1_base64="HjV70tkeR6sIrNLGcwZd9iqouBI=">AAAB+3icbZDNSsNAFIVv/K31L9alm8EiuNCSFEGXRTcuK5i2kIYwmU7aoZNJmJlIS+mruHGhiFtfxJ1v47TNQlsPDHycey/3zokyzpR2nG9rbX1jc2u7tFPe3ds/OLSPKi2V5pJQj6Q8lZ0IK8qZoJ5mmtNOJilOIk7b0fBuVm8/UalYKh71OKNBgvuCxYxgbazQroxChrpMIP/SC90LL6wHoV11as5caBXcAqpQqBnaX91eSvKECk04Vsp3nUwHEyw1I5xOy91c0QyTIe5T36DACVXBZH77FJ0Zp4fiVJonNJq7vycmOFFqnESmM8F6oJZrM/O/mp/r+CaYMJHlmgqyWBTnHOkUzYJAPSYp0XxsABPJzK2IDLDERJu4yiYEd/nLq9Cq11zDD1fVxm0RRwlO4BTOwYVraMA9NMEDAiN4hld4s6bWi/VufSxa16xi5hj+yPr8Abwfku4=</latexit><latexit sha1_base64="HjV70tkeR6sIrNLGcwZd9iqouBI=">AAAB+3icbZDNSsNAFIVv/K31L9alm8EiuNCSFEGXRTcuK5i2kIYwmU7aoZNJmJlIS+mruHGhiFtfxJ1v47TNQlsPDHycey/3zokyzpR2nG9rbX1jc2u7tFPe3ds/OLSPKi2V5pJQj6Q8lZ0IK8qZoJ5mmtNOJilOIk7b0fBuVm8/UalYKh71OKNBgvuCxYxgbazQroxChrpMIP/SC90LL6wHoV11as5caBXcAqpQqBnaX91eSvKECk04Vsp3nUwHEyw1I5xOy91c0QyTIe5T36DACVXBZH77FJ0Zp4fiVJonNJq7vycmOFFqnESmM8F6oJZrM/O/mp/r+CaYMJHlmgqyWBTnHOkUzYJAPSYp0XxsABPJzK2IDLDERJu4yiYEd/nLq9Cq11zDD1fVxm0RRwlO4BTOwYVraMA9NMEDAiN4hld4s6bWi/VufSxa16xi5hj+yPr8Abwfku4=</latexit><latexit sha1_base64="HjV70tkeR6sIrNLGcwZd9iqouBI=">AAAB+3icbZDNSsNAFIVv/K31L9alm8EiuNCSFEGXRTcuK5i2kIYwmU7aoZNJmJlIS+mruHGhiFtfxJ1v47TNQlsPDHycey/3zokyzpR2nG9rbX1jc2u7tFPe3ds/OLSPKi2V5pJQj6Q8lZ0IK8qZoJ5mmtNOJilOIk7b0fBuVm8/UalYKh71OKNBgvuCxYxgbazQroxChrpMIP/SC90LL6wHoV11as5caBXcAqpQqBnaX91eSvKECk04Vsp3nUwHEyw1I5xOy91c0QyTIe5T36DACVXBZH77FJ0Zp4fiVJonNJq7vycmOFFqnESmM8F6oJZrM/O/mp/r+CaYMJHlmgqyWBTnHOkUzYJAPSYp0XxsABPJzK2IDLDERJu4yiYEd/nLq9Cq11zDD1fVxm0RRwlO4BTOwYVraMA9NMEDAiN4hld4s6bWi/VufSxa16xi5hj+yPr8Abwfku4=</latexit>

Term-wise convex hull representation

2666664x1

x2

y

3777775= �1

2666664`1

`2

`1`2

3777775+ �2

2666664`1

u2

`1u2

3777775+ �3

2666664u1

`2

u1`2

3777775+ �4

2666664u1

u2

u1u2

3777775�1 + �2 + �3 + �4 = 1 and �

i

> 0 [i 2 {1, 2, 3, 4}<latexit sha1_base64="mVvu7U3DahLQFklBF3T9bspSODA=">AAAFQnichVRNb9NAEHVDgGK+WjhyWVGBEI1C7BZRkCpVQiCORTS0Um1F4/XYWWW9dr1riLF8hV/Gj+A3cAFx5cA6dZPaobCS5ad5M2/fzo7tJZxJNRh8W+lc6l6+cnX1mnn9xs1bt9fW77yXcZZSHNKYx+mRBxI5EzhUTHE8SlKEyON46E1eVvzhB0wli8WByhN0IwgFCxgFpUOj9ZUfjochE0XAgbNQPC7Nh6QOeRGolE1LczqyiOOQ6ciuXrnpoPDnLNklpsP1jj5Uac1S4iDnp9UzZM+RNQ80xDbnWvY/tLKGUPYPla0llaxtJ/u/l+2LVLKFxLIL3Uin6ua8OZtkcbgF3jqHt3U3tfBJBj5xFE5VAcIvzwJ1EiNOiCeSg1BkcMYFcQqcE80xQZzC6tm9rd62o12cXtf8fkdrG4P+YLbIMrBqsGHUa3+03nnh+DHNIhSKcpDy2Bokyi0gVYxyLE0nk5gAnUCIxxoKiFC6xWw2S/JAR3yi3elH+51Fz1cUEEmZR57O1H0byzZXBf/KeVFz5wPLLaotUNCWJRXsuAUTSTbjSIOUgY8BZFz1JAWO/m5/56lbvGYpvAMhG0KFwI9qKoOyOlKQcaJiUn1SxGcpUsVzDYCmTHeF0DGkQJX+8BoKSRgkPFYtXeBhrMvGEWs5dxeU3exzodjkU3mR9hmUqPQ1RAmoXe0dpaorapozL4U0LzRPx3Vts9gkBPXfIycwZfJJH5JEjxKRKueoB7VQY0YnZU8feNIK9jwOFVVNm9WerWUwtPvP+9Zbe2PvVT12q8Y9477xyLCMZ8ae8cbYN4YG7Rx2ys7nzpfu1+737s/ur9PUzkpdc9dorO7vPzIbxE0=</latexit><latexit sha1_base64="mVvu7U3DahLQFklBF3T9bspSODA=">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</latexit><latexit sha1_base64="mVvu7U3DahLQFklBF3T9bspSODA=">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</latexit><latexit sha1_base64="mVvu7U3DahLQFklBF3T9bspSODA=">AAAFQnichVRNb9NAEHVDgGK+WjhyWVGBEI1C7BZRkCpVQiCORTS0Um1F4/XYWWW9dr1riLF8hV/Gj+A3cAFx5cA6dZPaobCS5ad5M2/fzo7tJZxJNRh8W+lc6l6+cnX1mnn9xs1bt9fW77yXcZZSHNKYx+mRBxI5EzhUTHE8SlKEyON46E1eVvzhB0wli8WByhN0IwgFCxgFpUOj9ZUfjochE0XAgbNQPC7Nh6QOeRGolE1LczqyiOOQ6ciuXrnpoPDnLNklpsP1jj5Uac1S4iDnp9UzZM+RNQ80xDbnWvY/tLKGUPYPla0llaxtJ/u/l+2LVLKFxLIL3Uin6ua8OZtkcbgF3jqHt3U3tfBJBj5xFE5VAcIvzwJ1EiNOiCeSg1BkcMYFcQqcE80xQZzC6tm9rd62o12cXtf8fkdrG4P+YLbIMrBqsGHUa3+03nnh+DHNIhSKcpDy2Bokyi0gVYxyLE0nk5gAnUCIxxoKiFC6xWw2S/JAR3yi3elH+51Fz1cUEEmZR57O1H0byzZXBf/KeVFz5wPLLaotUNCWJRXsuAUTSTbjSIOUgY8BZFz1JAWO/m5/56lbvGYpvAMhG0KFwI9qKoOyOlKQcaJiUn1SxGcpUsVzDYCmTHeF0DGkQJX+8BoKSRgkPFYtXeBhrMvGEWs5dxeU3exzodjkU3mR9hmUqPQ1RAmoXe0dpaorapozL4U0LzRPx3Vts9gkBPXfIycwZfJJH5JEjxKRKueoB7VQY0YnZU8feNIK9jwOFVVNm9WerWUwtPvP+9Zbe2PvVT12q8Y9477xyLCMZ8ae8cbYN4YG7Rx2ys7nzpfu1+737s/ur9PUzkpdc9dorO7vPzIbxE0=</latexit>

V-representation (Bilinear)

�10

1 �1 �0.5 0 0.5 1

�1

0

1

x1x2

x

1x

2

Convex Hull of Bilinear Function

<latexit sha1_base64="+8KTH6/dn8m38tEx0hadQ9bBXBI=">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</latexit><latexit sha1_base64="+8KTH6/dn8m38tEx0hadQ9bBXBI=">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</latexit><latexit sha1_base64="+8KTH6/dn8m38tEx0hadQ9bBXBI=">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</latexit>

V-representation (Trilinear)

’k=1..8

k

= 1,

k

> 0, [k = 1, . . . , 8,

bx =’k=1..8

k

�(⇠k),

x

i

=’k=1..8

k

⇠ik<latexit sha1_base64="PwvW+OBWU4q7ymAWVxk4L24foLI=">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</latexit><latexit sha1_base64="PwvW+OBWU4q7ymAWVxk4L24foLI=">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</latexit><latexit sha1_base64="PwvW+OBWU4q7ymAWVxk4L24foLI=">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</latexit><latexit sha1_base64="PwvW+OBWU4q7ymAWVxk4L24foLI=">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</latexit>

�(x1, x2, x3) = x1x2x3, x

i

6 x

i

6 x

i

[i = 1, 2, 3<latexit sha1_base64="HIRYaY3ccJ1QiNMEZSSMwoHc5GY=">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</latexit><latexit sha1_base64="KJgYtouKpib/oDio/l4ezSOrLJo=">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</latexit><latexit sha1_base64="KJgYtouKpib/oDio/l4ezSOrLJo=">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</latexit><latexit sha1_base64="kYBroYhEH1WWByVedQeI9/SOT+Y=">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</latexit>

Meyer, C.A. and Floudas, C.A. “Trilinear monomials with mixed sign domains: Facets of the convex and concave envelopes” Journal of Global Optimization - 2004H-representation

(Trilinear)

Convex combination of extreme points

Improved QC-relaxation gaps

Without Bound Tightening

Instances QCrmc (�) QCconv (�)

case�_lmbd �.�� �.��case��_ieee ��.�� ��.��

case�_lmbd_api �.�� �.��case��_ieee_rts_api ��.�� �.��case��_ieee_rts_api ��.�� �.��

case�_lmbd_sad �.�� �.��case�_gs_sad �.�� �.��case�_pjm_sad �.�� �.��case��_ieee_rts_sad �.�� �.��case��_ieee_rts_sad �.�� �.��case���_ieee_sad �.�� �.��

<latexit sha1_base64="9YK39IOmYuxdxF2WdvvakNqMK8k=">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</latexit><latexit sha1_base64="9YK39IOmYuxdxF2WdvvakNqMK8k=">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</latexit><latexit sha1_base64="9YK39IOmYuxdxF2WdvvakNqMK8k=">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</latexit><latexit sha1_base64="9YK39IOmYuxdxF2WdvvakNqMK8k=">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</latexit>

Instances: C. Coffrin et. al, “NESTA, the NICTA energy system test archive,” 2014

Comparison of trilinear envelopes on OPF relaxations

MediumIPOPT

LargeIPOPT

MediumGUROBI

LargeGUROBI

MediumCPLEX

LargeCPLEX

101

102

103

Com

puta

tion

tim

e(s

econ

ds)

Recursive MC

Extreme point

Meyer-Floudas

Narimani, M.R, Molzahn, D.K, Nagarajan, H, Crow, M.L. “Comparison of Various Trilinear Monomial Envelopes for Convex Relaxations of Optimal Power Flow Problems”. IEEE Global Conference on Signal and Information Processing (GlobalSIP). IEEE, 2018.

0.000

2.000

4.000

6.000

8.000

10.000

12.000

14.000

16.000

18.000

20.000

case5_pjm

case240_wecc

case30_fsr

_api

case73_iee

e_rts_ap

i

case89_pegase

_api

case118_ieee_api

case189_edin_api

case29_edin_sad

case9_na_cao

_nco

case9_nb_cao_nco

case14_s_cao_nco

Without Obj upper bnd With Obj upper bnd (OBBT)

With Optimization-based Bound Tightening (OBBT)

Improved QC-relaxation gaps

X

i2Gc2i(R(Sg

i )2) + c1iR(Sg

i ) + c0i 6 UBfeas

<latexit sha1_base64="IMm8Ng+HrkcZ91a8T+AjIYccF8w=">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</latexit><latexit sha1_base64="AtFxJ/xVU17MAq/aLnG0zRgSf60=">AAACo3icbVHdatswGJW9remyv7S9227EQiFlEOzcbJelLaywm66r20LsGFmWUxH9uJK8EoTepW9VdrFHGUyOy2jafSB0+M75pKOjomZUmyj6FYTPnr/Y6G2+7L96/ebtu8HW9rmWjcIkwZJJdVkgTRgVJDHUMHJZK4J4wchFsThs+YufRGkqxZlZ1iTjaC5oRTEyvpUPrlPd8NzSlAqYcmSuMGL2q3MwLbTFLrcT6kZppeypG/3I6Wy+N5vswU//6Ji6NfYBFVF/CiPXmiFhYHIwsxVB2uWDYTSOVgWfgvgeDPePb3u/j3aSk3wreJ+WEjecCIMZ0noaR7XJLFKGYkZcP200qRFeoDmZeigQJzqzq2wc3PWdElZS+eVtrLoPJyziWi954ZXt8/Vjrm3+j5s2pvqSWSrqxhCBu4uqhkEjYRs0LKki2LClBwgr6r1CfIUUwsZ/Rz8V5AZLzpEorQ/MTePM75KVrRfJ7DB2bl1UqU7UOqkUWnQSn2b8OLun4HwyjqNx/N3HegC62gQfwEcwAjH4DPbBMTgBCcDgDvwJNoJeuBt+C0/Ds04aBvczO2CtwuwvXr3TAg==</latexit><latexit sha1_base64="AtFxJ/xVU17MAq/aLnG0zRgSf60=">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</latexit><latexit sha1_base64="Y12SWHrU51MoKD1x5V2C0SEb4ys=">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</latexit>

Local-solver (Ipopt)

Global optimum - 80% of hard instances

Global optimization: Tight piecewise relaxations

�10

1 �1 �0.5 0 0.5 1

�1

0

1

x1x2

x

1x

2

Bilinear Function

<latexit sha1_base64="LQM/oczlqDU3l4eb8elSolpN8Ag=">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</latexit><latexit sha1_base64="LQM/oczlqDU3l4eb8elSolpN8Ag=">AAAKRXicnVXNbttGEGZSpY3Un8TNrb2sqwiwA0omaRh1YxAIWjToMUXjJoAjGEtyKS28/MHu0iZN8NCn6bV9i75CX6Knotd2dpf6oeTYQAkR4sy38803u8NhkDMqpOP8ee/+B70HH370sD/4+JNPP3v0eOfzn0VW8JCchhnL+NsAC8JoSk4llYy8zTnBScDIm+DiO4W/uSRc0Cx9LaucTBM8S2lMQyzBdb7T++JdPotzlkkhiKw1YYJz5KOak6gxjj0nTPb9PXDsekf7J8g43XUnalAzeC/VjBOSdsm0a5tu6b6dMGA4vOgSateu52wQLt2acPAuIDOa1pJeXOc0lAUnzcKHSyqaswGCS2+kX39L1bZijl4Waaj2q7GRxkuGA8L8p+W5+9RG1dLywLrWFtKg9kCESDBj9iUlV3597DS16wDTjNPIRqPDCAVZCYuqhKb+2AW+BJe+C54p6MVRpHbg8EwUPLaRwEnOiPBdx0bLDUlxQnxdqI2iLMGa57m7MFDlO8/dKarLZ1VzskVpdPxPYjCdFXPE8dWZjjZRU7QHi8f63kfjsTbVzxhjY43XzPEKXUV2maHjDC8odVaR2joZjG5aOla/JesNcTfnN2EbnAEryGb+u6rqrGyFtjWpDkNXNJJz30uSdu+V8N0jZ6V94q2Ha/vkVgKlcsEAIR0Gd8kAIkgamc43j+svxvnjoTNx9IW2H9z2YWi116vznQdP3kVZWCQklSHDQpy5Ti6nNeaShky9aYUgOfQFnpEzeFTtJabmLW7QCDwRijMOdypNHesRNU6EqJIAViZYzsUmppwLrJOqDpJu6tfutFY5SBpuaJLx8bSmaV5oDHVAEUckxgWTtggxI5E/OT6a1i8pxz/hdCNjSq5kKeJG1RQXDMkMqemLIspJKFkFDzjkFLYFhXPMcShhRncYFpOvy4vZLIOweUI3lE9XkNfdaH2kzfu4uwM2ybH0QTsRUs3KdRDGEYEPSYVUsxxMcJ5DuyAhK0bUQJZzCgPZhoIuNpy2mdVqDI7UjYYuYCWMORvBWV8bnwc+OSdIzGks0SWG9gXgQGk/SNUwYgY6mIRZRJCHMJ8JlWOkB7KSKjkcA7RPohe2wALMM5pKfW710GvqZWnan8Fxc5PlEo4n41msCtGDXJVbD92mMXz6r7FHbTWQckYOYw6NfDBZ1t1+Q2ArMKOzdKHDR5BI0IjYeoHpIr1OLBbEmIkWJimDGuHDnlAptuGlwHbLFFyr160xuNKN1GwQz5bpGYmlQfXQCGHCo5U4nhVpZOCQ0TVoK3e5dswLuF4/Jb8uGyOmVVPdHVF1I67vjrjuRpSrmlcteLuquyM2Vd0d0VHVDJoBjFF3c2huP5x6k28m7o/e8MX37Tx9aH1pfWXtWa71tfXC+sF6ZZ1aYe+X3q+933q/9//o/9X/u/+PWXr/XhvzxOpc/X//AzOjKG0=</latexit><latexit sha1_base64="LQM/oczlqDU3l4eb8elSolpN8Ag=">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</latexit>

Tight piecewise formulations

�10

1 �1 �0.5 0 0.5 1

�1

0

1

x1x2

x

1x

2

Piecewise Mccormick Envelopes

<latexit sha1_base64="bupHhtFG0xTlAgPsuctbV5IWLnE=">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 AtS8xP82iYIKM+mKHVxm8laAW5a2SVxdHGb5VEkx+8m0lQ3VH6sMYdCNXcG0VRc57orv3yubNp3OFou3eMfPFWAt2iKwZ0aTG4awYMgsVhfbbXy+2j/+bxqTN0zID9hdssTq1mXN6cHJ1PwoTmEYsVFUTKa 9dJlV+STHEq9Lskx6LBo47M2DUu9ZEp/dJEXEEfkRCmSYYTS8Gg2x4liaQsogAtI6LmclenwYO6IGrf+ZXrl/oWLKY7IanpmV/yOM2NDlpKOQ3ZlORC2ZISwUJvePbcL3/gGfmFxLJFVMZsoZZyWulHmuYCVGIaA0KeMapEgQtCM467AnROMkIVvpZbDOlsqrtvh5eIWYJuc3zXtiP3N6pRe59NRqt/4l4tJVOYhgi70sPYmVSNR6MWPMhIVpSmaxvftjNWNcNuL0DX0rMhSVOsJpCqwI7yoFRz/DqobKW/Edqgbd59la2rzd2trf3F1Wj43dD9eXT64mVTdsfWF9ZX1lPLtb61Xlg/WpfWlUU7eee3zu+dP47/wlP+UfekNj160Ph8ZrVG9/O/AWAxrKA=</latexit><latexit sha1_base64="bupHhtFG0xTlAgPsuctbV5IWLnE=">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 AtS8xP82iYIKM+mKHVxm8laAW5a2SVxdHGb5VEkx+8m0lQ3VH6sMYdCNXcG0VRc57orv3yubNp3OFou3eMfPFWAt2iKwZ0aTG4awYMgsVhfbbXy+2j/+bxqTN0zID9hdssTq1mXN6cHJ1PwoTmEYsVFUTKa 9dJlV+STHEq9Lskx6LBo47M2DUu9ZEp/dJEXEEfkRCmSYYTS8Gg2x4liaQsogAtI6LmclenwYO6IGrf+ZXrl/oWLKY7IanpmV/yOM2NDlpKOQ3ZlORC2ZISwUJvePbcL3/gGfmFxLJFVMZsoZZyWulHmuYCVGIaA0KeMapEgQtCM467AnROMkIVvpZbDOlsqrtvh5eIWYJuc3zXtiP3N6pRe59NRqt/4l4tJVOYhgi70sPYmVSNR6MWPMhIVpSmaxvftjNWNcNuL0DX0rMhSVOsJpCqwI7yoFRz/DqobKW/Edqgbd59la2rzd2trf3F1Wj43dD9eXT64mVTdsfWF9ZX1lPLtb61Xlg/WpfWlUU7eee3zu+dP47/wlP+UfekNj160Ph8ZrVG9/O/AWAxrKA=</latexit><latexit sha1_base64="bupHhtFG0xTlAgPsuctbV5IWLnE=">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 AtS8xP82iYIKM+mKHVxm8laAW5a2SVxdHGb5VEkx+8m0lQ3VH6sMYdCNXcG0VRc57orv3yubNp3OFou3eMfPFWAt2iKwZ0aTG4awYMgsVhfbbXy+2j/+bxqTN0zID9hdssTq1mXN6cHJ1PwoTmEYsVFUTKa 9dJlV+STHEq9Lskx6LBo47M2DUu9ZEp/dJEXEEfkRCmSYYTS8Gg2x4liaQsogAtI6LmclenwYO6IGrf+ZXrl/oWLKY7IanpmV/yOM2NDlpKOQ3ZlORC2ZISwUJvePbcL3/gGfmFxLJFVMZsoZZyWulHmuYCVGIaA0KeMapEgQtCM467AnROMkIVvpZbDOlsqrtvh5eIWYJuc3zXtiP3N6pRe59NRqt/4l4tJVOYhgi70sPYmVSNR6MWPMhIVpSmaxvftjNWNcNuL0DX0rMhSVOsJpCqwI7yoFRz/DqobKW/Edqgbd59la2rzd2trf3F1Wj43dD9eXT64mVTdsfWF9ZX1lPLtb61Xlg/WpfWlUU7eee3zu+dP47/wlP+UfekNj160Ph8ZrVG9/O/AWAxrKA=</latexit>

z1 = 1) �1 6 x2 6 0

2666664x1

x2

y

3777775= �1

2666664�1�11

3777775+ �2

2666664�100

3777775+ �3

26666641�1�1

3777775+ �4

2666664100

3777775�1 + �2 + �3 + �4 = 1 and �5 + �6 = 0

<latexit sha1_base64="YeHuCzsW9w8GIborAVXpgQXQId8=">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</latexit><latexit sha1_base64="YeHuCzsW9w8GIborAVXpgQXQId8=">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</latexit><latexit sha1_base64="YeHuCzsW9w8GIborAVXpgQXQId8=">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</latexit><latexit sha1_base64="YeHuCzsW9w8GIborAVXpgQXQId8=">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</latexit>

z2 = 1) 0 6 x2 6 1

2666664x1

x2

y

3777775= �2

2666664�100

3777775+ �5

2666664�11�1

3777775+ �4

2666664100

3777775+ �6

2666664111

3777775�2 + �4 + �5 + �6 = 1 and �1 + �3 = 0

<latexit sha1_base64="UrjVuoPt+IoQpjJgs0kEOMH/T4k=">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</latexit><latexit sha1_base64="UrjVuoPt+IoQpjJgs0kEOMH/T4k=">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</latexit><latexit sha1_base64="UrjVuoPt+IoQpjJgs0kEOMH/T4k=">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</latexit><latexit sha1_base64="UrjVuoPt+IoQpjJgs0kEOMH/T4k=">AAAFIHicdZTfb9MwEMezUGCUXxs88mKxISE2uqTQMZAmTUIgHgesbNISVRfHSa06TmY7rFmUv4a/hhcQj/DX4KTp2nSrpViX+/rjO58v8RJGpbKsvyvmjdbNW7dX77Tv3rv/4OHa+qNvMk4FJn0cs1iceCAJo5z0FVWMnCSCQOQxcuyN3pf68XciJI35kcoS4kYQchpQDEq7BusrvxxMuCKC8hBtXgy6aB/ZyPlCw6ECIeJzZCGHkTPJgCs01vrszd5sOx4JKc8DBoyG/EUxdXgRKEHHRVsjejtnQjooazuE+5eyDtZ2mM7Wh1JvsuhlRVrV1MS2LqneEqqaSnMJ9/oKNxdsKbV7PTWdGpTjzB1tC80Cz+zenL07KfxZCj5yFBmrHAH3UTF11cvsOeSVRqxJQS8vYLC2YXWsaqCrhl0bG0Y9Dgfr5jvHj3Ea6S7ADKQ8ta1EuTkIRTEj+kZTSRLAIwjJqTY5RES6edV5BXqmPT4KYqEf3RGVd57IIZIyizy9UtdlKBe10nmt5kXNyEe2m5chCMcLKalgz80pT9JKQw1RBj4JIGVqW2JgxN/v7PXc/CMV8BW4bGyUc3KuxjIoyiMFKUMqRuUHg3wqCFYs0wZgQXVVEB6CAKw/mmbOSRgkLFYL+wILY40NI7qQuTuTus0654qOLople09NSZS+higBta9zJ1LVRC0z6gkQWa51PKzZJtxGiOh/Q4ZgTOVOB5JEtxKSKmNEd1auhhSPim194NGCc9tjUEplt9mLvXXV6Hc7bzv25+7GwYe67VaNJ8ZT47lhG2+MA+OTcWj0DWz2zFPTN0nrR+tn63frz2SpuVIzj43GaP37D9EQtWI=</latexit>

z1 + z2 = 1, z1, z2 2 {0, 1}, and �i > 0 i 2 {1, . . . , 6}<latexit sha1_base64="NPki55L2RfWY/GM1G/XGve+0lds=">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</latexit><latexit sha1_base64="NPki55L2RfWY/GM1G/XGve+0lds=">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</latexit><latexit sha1_base64="NPki55L2RfWY/GM1G/XGve+0lds=">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</latexit><latexit sha1_base64="NPki55L2RfWY/GM1G/XGve+0lds=">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</latexit>

Tight piecewise formulations

�10

1 �1 �0.5 0 0.5 1

�1

0

1

x1x2

x

1x

2

Bilinear Function

<latexit sha1_base64="82upMd9CD/stVMAGfo938mI7qwA=">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</latexit><latexit sha1_base64="82upMd9CD/stVMAGfo938mI7qwA=">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</latexit><latexit sha1_base64="82upMd9CD/stVMAGfo938mI7qwA=">AAAK2nicnVZfb9s2EFe7euu0f80K7GUvzFwDSSE7koJgWTMBxYYGe+ywZi2QGAEtUTYR6g9IOpEi6GlPw173PfZ59m12JGVZstdkmGBBvPvxfrw7Ho+e5YwK6bp/P3j4waPBhx89/tj+5NPPPv/iyc6Xv4psyUNyFmYs4+9mWBBGU3ImqWTkXc4JTmaMvJ1d/ajwt9eEC5qlb2SZk2mC5ymNaYglqC53Hv11kc/jnGVSCCIrTZjgHAWo4iSqjWLPDZP9YA8Uu/7R/gkySq+rRDWq7fdSzTkhaZ9Mq7bpWvXdhDOGw6s+oVbt+u4GYavWhPbFjMxpWkl6dZvTUC45qVc6XFBRn9sIHp3IoPqBqrRijk6XaajyVTtI4wXDM8KCZ8Wl98xBZSv5IN1qCWlQa8BCJJgx55qSm6A6duvKc4FpzmnkoNFhhGZZAZPKhKbB2AO+BBeBB5op+IujSGXg8FwseewggZOcERF4roPahKQ4IYEO1EFRlmDN88JbCagM3BfeFFXF87I+2aI0fvxPYhDdNXPE8c25tjZWU7QHk8f63UfjsRbVzwhjI4074niNri37zFBxDa878Y+aT4fBaLVy06UlMZadKa1dl+zEHv3bimP1a52DRLkd857UC8OYbXB2nGkt70tOb6YWTuwmQlWo6IZGchH4SdJsoXJ898hd+z7xW5+acJXq5E6OjON0Tloabddl+s9EKuIVDdqiMbFqEvuCpJE5jmbYPa2XT4buxNUP2h54zWBoNc/ry53B04soC5cJSWXIsBDnnpvLaYW5pCFTx38pSA7FiufkHIaq5sXUtJYajUAToTjj8KbSRNK1qHAiRJnMYGaC5UJsYkq5wnpLVbOkv/Qbb1qpNUgabvgk4+NpRdN8qTHUA0UckRgvmXREiBmJgsnx0bQ6pRz/gtONFVNyIwsR1yqmeMmQzJC6ElBEOQklK2GAQ04hLShcYI5DCRdHj2HVjvu8mM0zMFskdMPz6Rry+4nWW1q/j7vf9ZMcywB8J0KqBt4FoUcSuN1KpIrlYILzHMoFCVkyom4JuaBwSzgQ0NWG0jEXiOrNI/WioQdYAb3XQbDXt0bng04uCBILGkt0jaGAAThQvh+kqkMyAx1MwiwiyEeYz4VaY6RvCeWqhNMjoHwSPbEBVmCe0VTqfauGfl21oWl9BtvNzSrXsD0Zz2IViL5dVLjV0Ktrw6c/tTNqotEH9jDmUMgHkzbu5mKDVGBG5+nKjwDBQoJGxNETTBXpeWI1IcZMNDBJGcQI/zYSKsU23DrYpEzBlTputcGV30h1B/G8XZ6RWBpUt40Qrh20do5nyzQycMhoB9pau+hs8wquursUVEVtnGm8Ke+3KPsWt/db3PYtinXM6xK826v7LTa9ut+i51Vt1za0UW+zaW4PzvzJdxPvZ3/48lXTTx9bX1vfWHuWZ31rvbR+sl5bZ1Y4+Grw/eDV4NSe2r/Zv9t/mKkPHzQ2T63eY//5D+yRSlo=</latexit>

�10

1 �1 �0.5 0 0.5 1

�1

0

1

x1x2

x

1x

2

Piecewise Mccormick Envelopes

<latexit sha1_base64="olly39wog8yVXfcs5qL//rl7HQo=">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</latexit><latexit sha1_base64="olly39wog8yVXfcs5qL//rl7HQo=">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</latexit><latexit sha1_base64="olly39wog8yVXfcs5qL//rl7HQo=">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</latexit>

Bivariate partitioning

�(x̂s

) = x1,i · x2,j

x1

x2

x̂1x̂2 x̂3

x̂4

x̂5x̂6 x̂7

x̂8

x̂9x̂10 x̂11

x̂12

x̂13x̂14 x̂15

x̂16

x1,1 x1,2 x1,3 x1,4

x2,1

x2,2

x2,3

x2,4

z

11 z

21 z

31

z

12

z

22

z

32

<latexit sha1_base64="9u1d+eEoKvqs0G8k2KFrzm3+n10=">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</latexit><latexit sha1_base64="9u1d+eEoKvqs0G8k2KFrzm3+n10=">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</latexit><latexit sha1_base64="9u1d+eEoKvqs0G8k2KFrzm3+n10=">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</latexit><latexit sha1_base64="9u1d+eEoKvqs0G8k2KFrzm3+n10=">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</latexit>

Tight piecewise formulations

SOS-2 type constraints: Extreme points of the lifted-variable polytope are integral

Tight piecewise formulations

Uniformly spaced partitions induce too many binaries. Hence, formulating tractable mixed-integer convex programs are crucial

Adaptive variable partitioning: Tightening gaps using mixed-integer convex programsLocal solvers (IPOPT) are amazing on ACOPF

�3 �2 �1 0 1 2 3

�2.5 6 x 6 2.5

Iterations

�3 �2 �1 0 1 2 3

x

⇤ Active partition

x

⇤ Active partition

�3 �2 �1 0 1 2 3

Non-uniform, dynamically added partitions guided by local and lower-bounding solutions

H. Nagarajan, M. Lu, E. Yamangil, R. Bent, “Tightening McCormick Relaxations for Nonlinear Programs via Dynamic Multivariate Partitioning,” Constraint Programming, 2016

H. Nagarajan, M. Lu, S. Wang, R. Bent, K. Sundar, “An Adaptive, Multivariate Partitioning Algorithm for Global Optimization of Nonconvex Programs,” Journal of Global Optimization, 2018

Adaptive variable partitioning

Was found useful in many applications

POD.jl: An open-source global MINLP solverhttps://github.com/lanl-ansi/POD.jl

POD.jl: An open-source global MINLP solver

���

��

BARON POD

SCIPCOUENNE

� gap remaining

�ofinstances

�.��� �.��� �� ��� ����

��

���

���

���

���

����

BARON

POD

COUENNE

SCIP

Numerical results on ACOPF

3 hours time limit for piecewise, dynamic

partitioning algorithm

0.000

2.000

4.000

6.000

8.000

10.000

12.000

14.000

16.000

18.000

20.000

case5_pjm

case240_wecc

case30_fsr

_api

case73_iee

e_rts_ap

i

case89_pegase

_api

case118_ieee_api

case189_edin_api

case29_edin_sad

case9_na_cao

_nco

case9_nb_cao_nco

case14_s_cao_nco

Without Obj upper bnd With Obj upper bnd (OBBT)

Global optimum - 80% of hard instances

Global optimum - 95% of hard instances (N <= 300)

Challenging Instances

case89_pegase_api (4.5%)case118_ieee_api (1.1%)case9_bgm_nco (3.5%)case39_1_bgm_nco (3.3%)

Revisiting extreme-point formulationInstances QC

rmc

(%) QC

conv

(%)

case3 lmbd 1.21 0.96

case30 ieee 15.64 15.20

case3 lmbd api 1.79 1.59

case24 ieee rts api 11.88 8.78

case73 ieee rts api 10.97 9.64

case3 lmbd sad 1.42 1.37

case4 gs sad 1.53 0.96

case5 pjm sad 0.99 0.77

case24 ieee rts sad 2.93 2.77

case73 ieee rts sad 2.53 2.38

case118 ieee sad 4.61 4.14

case179 goc api 7.18 7.21

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Unexpected

Revisiting extreme-point formulation

´gijvivj qcsij ´ bijvivj|snij<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Extreme point captures the convex hull locally on trilinear

monomials

´gijvivj qcsij ´ bijvivj|snij<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Recursive McCormick shares the lifted variable across the trilinear

monomials

´gijvivj qcsij ´ bijvivj|snij<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

But neither captures the convex hull of

Tying constraint

'pvi, vj , qcsij , |snijq “ ´gij vivj qcsijlooomooon

�c

´bij vivj|snijlooomooon

�s<latexit sha1_base64="Hdggllwa4zy3Y4qG1H2U9Rk6UMk=">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</latexit><latexit sha1_base64="Hdggllwa4zy3Y4qG1H2U9Rk6UMk=">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</latexit><latexit sha1_base64="Hdggllwa4zy3Y4qG1H2U9Rk6UMk=">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</latexit><latexit sha1_base64="Hdggllwa4zy3Y4qG1H2U9Rk6UMk=">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</latexit>

Computational results on bound tightening with improved relaxationshttps://arxiv.org/abs/1809.04565

Revisiting extreme-point formulation

Revisiting extreme-point formulation

Instances QC

rmc

(%) QC

conv

(%)

case3 lmbd 1.21 0.96

case30 ieee 15.64 15.20

case3 lmbd api 1.79 1.59

case24 ieee rts api 11.88 8.78

case73 ieee rts api 10.97 9.64

case3 lmbd sad 1.42 1.37

case4 gs sad 1.53 0.96

case5 pjm sad 0.99 0.77

case24 ieee rts sad 2.93 2.77

case73 ieee rts sad 2.53 2.38

case118 ieee sad 4.61 4.14

case179 goc api 7.18 7.21

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

6.98%<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

QC-RMC

QC-LMAC

QC-TLM

https://arxiv.org/abs/1809.04565

Future directions

✦ Extension of tying constraints to the partitioned case

✦ Other OPF formulations (Rectangular, Current-Voltage-based, Tangent)

✦ Incorporating SDP-based cuts by exploiting graph sparsity

✦ Scaling to large-scale networks?

Thank you!

Questions?

top related