lecture 6: interior point method...motivation • simplex method works well in general, but suffers...
TRANSCRIPT
![Page 1: LECTURE 6: INTERIOR POINT METHOD...Motivation • Simplex method works well in general, but suffers from exponential-time computational complexity. • Klee-Minty example shows simplex](https://reader036.vdocument.in/reader036/viewer/2022090810/611c48edff9d7c29fa439447/html5/thumbnails/1.jpg)
LECTURE 6: INTERIOR POINT METHOD1. Motivation2. Basic concepts3. Primal affine scaling algorithm4. Dual affine scaling algorithm
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Motivation• Simplex method works well in general, but suffers from
exponential-time computational complexity.• Klee-Minty example shows simplex method may have to
visit every vertex to reach the optimal one.• Total complexity of an iterative algorithm
= # of iterations x # of operations in each iteration• Simplex method
- Simple operations: Only check adjacent extreme points- May take many iterations: Klee-Minty example
Question: any fix?
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Complexity of the simplex method
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Worst case performance of the simplex methodKlee-Minty Example:• Victor Klee, George J. Minty, “How good is the simplex
algorithm?’’ in (O. Shisha edited) Inequalities, Vol. III (1972), pp. 159-175.
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Klee-Minty Example
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Klee-Minty Example
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Karmarkar’s (interior point) approach• Basic idea: approach optimal solutions from the interior ofthe feasible domain
• Take more complicated operations in each iteration to find a better moving direction
• Require much fewer iterations
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General scheme of an interior point method
• An iterative method that moves inthe interior of the feasible domain
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Interior movement (iteration)• Given a current interior feasible solution , we have
> 0
An interior movement has a general format
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Key knowledge• 1. Who is in the interior?
- Initial solution
• 2. How do we know a current solution is optimal?- Optimality condition
• 3. How to move to a new solution?- Which direction to move? (good feasible direction)- How far to go? (step-length)
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Q1 - Who is in the interior?• Standard for LP
• Who is at the vertex?• Who is on the edge?• Who is on the boundary?• Who is in the interior?
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What have learned before
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Who is in the interior?• Two criteria for a point x to be an interior feasible solution:
1. Ax = b (every linear constraint is satisfied)2. x > 0 (every component is positive)
• Comments:1. On a hyperplane ,
every point is interior relative to H.2. For the first orthant K = ,
only those x > 0 are interior relative to K.
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Example
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How to find an initial interior solution?• Like the simplex method, we have
- Big M method- Two-phase method
(to be discussed later!)
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Key knowledge• 1. Who is in the interior?
- Initial solution
• 2. How do we know a current solution is optimal?- Optimality condition
• 3. How to move to a new solution?- Which direction to move? (good feasible direction)- How far to go? (step-length)
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Q2 - How do we know a current solution is optimal?
• Basic concept of optimality:A current feasible solution is optimal if and only if “no feasible direction at this point is a good direction.”
• In other words, “every feasible direction is not a good direction to move!”
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Feasible direction• In an interior-point method, a feasible direction at a current solution is a direction that allows it to take a small movement while staying to be interior feasible.
• Observations:
- There is no problem to stay interior if the step-length issmall enough.
- To maintain feasibility, we need
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Good direction• In an interior-point method, a good direction at a current solution is a direction that leads it to a new solution with a lower objective value.
• Observations:
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Optimality check• Principle:
“no feasible direction at this point is a good direction.”
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Key knowledge• 1. Who is in the interior?
- Initial solution
• 2. How do we know a current solution is optimal?- Optimality condition
• 3. How to move to a new solution?- Which direction to move? (good feasible direction)- How far to go? (step-length)
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Q3 – How to move to a new solution?1. Which direction to move?
- a good, feasible direction“Good” requires
“Feasible” requires
Question: any suggestion?
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A good feasible direction• Reduce the objective value
• Maintain feasibility
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Projection mapping• A projection mapping projects the negative gradient vector
–c into the null space of matrix A
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Q3 – How to move to a new solution?2. How far to go?
- To satisfy every linear constraintSince
the step-length can be real number.
- To stay to be an interior solution, we need
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How to choose step-length?• One easy approach
- in order to keep > 0
we may use the “minimum ratio test” to determine the step-length.
Observation: - when is close to the boundary, the step-length
may be very small.Question: then what?
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Observations• If a current solution is near the center of the feasible
domain (polyhedral set), in average we can make adecently long move.
• If a current solution is not near the center, we need tore-scale its coordinates to transform it to become “near the center".
Question: but how?
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Where is the center?• We need to know where is the “center” of the
non-negative/first orthant .-Concept of equal distanceto the boundary
Question: If not,what to do?
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Concept of scaling• Scale• Define a diagonal scaling matrix
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Transformation – affine scaling• Affine scaling transformation
• The transformation is1. one-to-one2. onto3. Invertible 4. boundary to boundary
5. interior to interior
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Transformed LP
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Step-length in the transformed space• Minimum ratio test in the y-space
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Property 1• Iteration in the x-space
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Property 2• Feasible direction in x-space
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Property 3• Good direction in x-space
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Property 4• Optimality check (Lemma 7.2)
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Property 5• Well-defined iteration sequence (Lemma 7.3)
From properties 3 and 4, if the standard form LPis bounded below and is not a constant, thenthe sequenceis well-defined and strictly decreasing.
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Property 6• Dual estimate, reduced cost and stopping ruleWe may define
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Property 7• Moving direction and reduced cost
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Primal affine scaling algorithm
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Example
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Example
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How to find an initial interior feasible solution?
• Big-M method Idea: add an artificialvariable with a big penalty
(big-M)
• Objective
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Properties of (big-M) problem(1) It is a standard form LP with n+1 variables and m
constraints.
(2) e is an interior feasible solution of (big-M).
(3)
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Two-phase method
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Properties of (Phase-I) problem
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Facts of the primal affine scaling algorithm
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More facts
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Improving performance – potential push• Idea: (Potential push method)
- Stay away from the boundary by adding apotential push. Consider
Use
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Improving performance – logarithmic barrier
• Idea: (Logarithmic barrier function method)Consider
Properties:
(2)
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Dual affine scaling algorithm• Affine scaling method applied to the dual LP
• Idea:
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Key knowledge• Dual scaling (centering) • Dual feasible direction• Dual good direction – increase the dual objective value• Dual step-length• Primal estimate for stopping rule
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Observation 1• Dual scaling (centering)
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Observation 2• Dual feasibility (feasible direction)
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Observation 3• Increase dual objective function (good direction)
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Observation 4• Dual step-length
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Observation 5• Primal estimateWe define
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Dual affine scaling algorithm
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Find an initial interior feasible solution
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Properties of (big-M) problem
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Performance of dual affine scaling• No polynomial-time proof !• Computational bottleneck
• Less sensitive to primal degeneracy and numerical errors,but sensitive to dual degeneracy.
• Improves dual objective value very fast, but attains primalfeasibility slowly.
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Improving performance1. Logarithmic barrier function method
Polynomial-time proof
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Improving performance• Power series method
- Basic idea: following a higher order trajectory
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Power series expansion