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A Non-Convex Combination of Gaussian Distributions

LINDO Systems has recently introduced LINDO API !" and LINGO #!"! $he ne% releases offera number of enhancements includin& si&nificantly ex'anded nonlinear ca'abilities and im'roved'erformance on linear and inte&er 'roblems!

Find Better Solutions to Tough Nonlinear Problems(any nonlinear models are nonconvex )e!&!* they have more than one local o'timum+ and* as aresult* not %ell suited to traditional solution techni,ues that rely on local search 'rocedures! Localsearch solvers are &enerally desi&ned to search only until they have identified a local o'timum! Ifthe model is non-convex* other local o'tima may exist that yield si&nificantly better solutions! $hene% Global solver and (ultistart feature included in the Nonlinear o'tion can hel' find bettersolutions to non-convex models!

Determine the Proven Global Optimumather than sto''in& after the first local o'timum is found* the Global solver %ill search until the&lobal o'timum is confirmed! $he Global solver converts the ori&inal non-convex* nonlinear'roblem into several convex* linear sub'roblems! $hen* it uses the branch-and-bound techni,ueto exhaustively search over these sub'roblems for the &lobal solution!

Use Multistart to Improve Solutions.hen limited time ma/es searchin& for the &lobal o'timum 'rohibitive* the ne% (ultistart featurecan be a 'o%erful tool for findin& &ood solutions more ,uic/ly! $his feature intelli&ently &eneratesa set of candidate startin& 'oints in the solution s'ace! $hen* the nonlinear solver intelli&entlyselects a subset of these to initiali0e a series of local o'timi0ations! 1or non-convex nonlinearmodels* the ,uality of the solution returned by the multistart solver %ill be su'erior to that of the

&eneral nonlinear solver!

Solve Linear and Integer Models Faster$he ne% releases of LINDO API and LINGO include enhancements that can si&nificantly im'rove'erformance on many linear and inte&er models!

Find Faster Solutions Using the Dual Solver$he im'roved Dual Sim'lex solver in the base versions of LINDO API !" and LINGO #!" deliverssubstantially better 'erformance! On broad classes of 'roblems* the ne% solver can 'rovide

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s'eed im'rovements of u' to 2""3! $he Dual Sim'lex solver is also even more robust due toim'roved handlin& of de&enerate and numerically unstable 'roblems!

Linearie Models !utomati"all#Nonlinear solvers that utili0e &radient based al&orithms do not 'erform %ell on models %ithnonsmooth functions! $he ne% Lineari0ation ca'abilities in LINDO API and LINGO can

dramatically im'rove 'erformance on models %ith common nonsmooth functions! $he featureautomatically converts the nonsmooth functions and o'erators to a series of linear*mathematically e,uivalent ex'ressions! (any nonsmooth models may be entirely lineari0ed! $hisallo%s the solver to ,uic/ly find a &lobal solution to %hat %ould have other%ise been anintractable 'roblem!

LINDO API can lineari0e the functions absolute value* if* and* or* not* max* and min as %ell as theo'erators 4* 45* 46* 5* 6* and 65! It can also reco&ni0e and lineari0e the 'roduct of a binary )"78+variable and a variable* or any 'roduct of t%o continuous or discrete variables constrained to bee,ual to* less than* or &reater than 0ero! LINGO functions su''orted by lineari0ation include9A:S)+* 9(A;)+* 9(IN)+* 9S(A;)+* and 9S(IN)+ alon& %ith any 'roducts of binary andcontinuous variables! $he feature also lineari0es all of LINGO

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include the :arrier o'tion can no% solvemodels in %hich the obective function and7orsome constraints include ,uadratic terms! :yta/in& advanta&e of the ,uadratic structure*LINGO can solve these models much more,uic/ly than usin& the &eneral nonlinear

solver! LINGO can even handle ,uadraticmodels %ith binary and &eneral inte&errestrictions! $hese ne% ,uadratic ca'abilitiesma/e LINGO suitable for a''lications such as'ortfolio o'timi0ation 'roblems* constrainedre&ression 'roblems* and certain classes ofdifficult lo&istics 'roblems )layout 'roblems*fixed-char&e-net%or/ 'roblems %ith ,uadraticobectives+!-Analy&e In'easi(le and )n(ounded!odels-- Determinin& %hy your model isinfeasible or unbounded no lon&er needs to bea dauntin& tas/! LINGO #!" includes a ne% set

of tools that allo% you to 'in'oint %hat iscausin& a model to be infeasible orunbounded! $he tools isolate a 'ortion of theori&inal model as the source of the 'roblem!$his allo%s you to focus your attention on arelatively small subsection of the model to loo/for formulation or data entry errors!-*uild !ulti-client and +e( A%%lications --LINGO #!" has been re-en&ineered to bethreadsafe for linear* inte&er* and ,uadraticmodels allo%in& one instance of the 'ro&ramto simultaneously %or/ on solvin& multi'lemodels! $his ma/es LINGO #!" ideal for multi-client and internet a''lications!

featured callable solver to offer &eneralnonlinear and nonlinear7inte&er ca'abilities!$his uni,ue feature allo%s develo'ers toincor'orate a nonlinear solver into their customa''lications! As %ith its linear and inte&erca'abilities* LINDO API 'rovides the user %ith

a com'rehensive set of routines for formulatin&*solvin&* and modifyin& nonlinear models! $heNonlinear o'tion is re,uired in order to utili0ethe nonlinear ca'abilities %ith LINDO API!-,un LIN"O A#I 'rom a +e( *roser --LINDO API !" includes ava Native Interface)NI+ su''ort for .indo%s* Solaris* and Linux'latforms! $his ne% feature allo%s users to callLINDO API in a''lets runnin& from a bro%ser!-E%anded !A/LA* Inter'ace -- $he (atlabinterface has been ex'anded to su''ort allLINDO API functions! Bsin& (A$LA: