Re: Constrained Min, non-linear

*To*: mathgroup at smc.vnet.net*Subject*: [mg4590] Re: Constrained Min, non-linear*From*: "Stas" <URYASEV at dnenet.nov.dne.bnl.gov>*Date*: Wed, 21 Aug 1996 03:25:13 -0400*Sender*: owner-wri-mathgroup at wolfram.com

>From: elsner at avalon.msfc.nasa.gov (Ron Elsner) >Subject: [mg4510] Re: Constrained Min, non-linear >Hi, >I'm not yet quite familiar with all Mathematica features (who is?) and I'm >looking for a solution of a rather simple problem. I can easily solve it by >hand but I wanna know how to do this with Mathematica. Here it is: >Where is f(x,y)=x^2+y^2 minimal, if x and y are >constrained to abs(x+y)=1 ? >Is it possible to solve this without programming? >Please answer via email, thanks in advance, Stefan. Stefan asks for an email response. I would also like to see the answer to this posted here. Ron Elsner NASA/MSFC ++++++++++++++++++++++++++++++++++++++++++++++++++++++++ As it formulated, this is an optimization problem with a nonsmooth constraint. There are a lot of publications in the area of nonsmooth optimization. For example, algorithms to solve nonsmooth problems are described in Uryasev S.P. " New Variable Metric Algorithms for Nondifferentiable Optimization Problems." Journal of Optimization Theory and Applications Vol. 71, No. 2, 359-388, 1991 . I have FORTRAN codes implementing the algorithms described in this paper. Recently, I translated one FORTRAN code to the MATHEMATICA language. I am planing to submit this code in MathSource. Implemented version of the algorithm solves unconstrained optimization problems. Therefore, I used a nonsmooth penalty function to include the constraint in the objective function. The problem has two local extremums. I started the algorithm from different initial approximations; algorithm converged to point (0.5,0.5) or (-0.5,-0.5) depending upon the initial approximation. This code needs two functions calculating the objective function and a generalized gradient. Further, I provide the codes of these functions with self explained identificators. I present two runs of the code leading to different local extremums. Stas Uryasev, Brookhaven National Laboratory, (516) 3447849 Needs["NonsmoothOptimization`VariableMetricAlgorithm`"]; fun[{x1_,x2_}]:= Module[{fn,PenaltyConstant,Constraint,ObjectiveFunction}, PenaltyConstant=10; Constraint=Sign[x1+x2]*(x1+x2) -1; ObjectiveFunction=x1^2+x2^2; fn= ObjectiveFunction + PenaltyConstant*Sign[Constraint]*Constraint; Return[fn] ]; grad[{x1_,x2_}]:= Module[{gr,PenaltyConstant,GrConstraint,GrObjectiveFunction, Constraint}, PenaltyConstant=10; Constraint=Sign[x1+x2]*(x1+x2) -1; GrConstraint={Sign[x1+x2],Sign[x1+x2]}; GrObjectiveFunction={2*x1,2*x2}; gr= GrObjectiveFunction + PenaltyConstant * Sign[Constraint] * GrConstraint; Return[gr] ]; x0={10,2}; VariableMetricAlgorithm[x0,NumberIterations -> 200, ArgumentPrecision->10^-20] VariableMetricAlgorithm::lstop3: Algorithm performed specified NumberIterations {0.5, {x1 -> 0.5, x2 -> 0.5}} x0={-10,-1}; VariableMetricAlgorithm[x0,NumberIterations -> 200, ArgumentPrecision->10^-20] VariableMetricAlgorithm::lstop3: Algorithm performed specified NumberIterations {0.5, {x1 -> -0.5, x2 -> -0.5}} ==== [MESSAGE SEPARATOR] ====