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Dual multipliers Description Nonlinear example for dual multipliers for non-linear problems Further explanation of this example:
'Xpress NonLinear Reference Manual'
Source Files xnlp_nlp_duals.mos ! XNLP example for dual multipliers for non-linear problems ! ! This example solves a series of problem, some of which do not have valid dual ! multipliers (Lagrange multipliers) in the absense of one of the regularity ! conditions (like Slater's condition) ! ! This example demonstrates a particular non-linear optimization concept as related ! to Xpress NonLinear. ! The version of the example is for Xpress 7.5. ! ! (c) 2013 Fair Isaac Corporation ! author: Zsolt Csizmadia model mmxnlp_nlp_duals uses "mmxnlp"; declarations x,y: mpvar end-declarations ! Simple model minimizing a linear objective over a ball Ball1 := x^2 + y^2 <= 1 x is_free y is_free ! Optimize with defaults. Note that y solution of y is not exact writeln("Step one: default options") minimize(y) writeln("Optimal solution: x=",getsol(x)," y=",getsol(y)) writeln("-----------------------------------------------") writeln ! Set a strict complementarity target. Bargaptarget is a "wish", it is safe to ! set it small, if the solver cannot satisfy it, it will stop early ! (in constrast to bargapstop) ! This is a solver specific setting, use the xprs_ pretag. setparam("xprs_bargaptarget",1e-20) minimize(y) writeln("Optimal solution: x=",getsol(x)," y=",getsol(y)) ! Look at the dual values ! Observe that the multipliers provide a proof of local optimality writeln("Optimal solution: pi(Ball1)=",getdual(Ball1)) writeln("Linearization of Ball1 at the optimal solution: (", 2*getsol(x), ",",2*getsol(y),")") writeln("Multiplied with its dual: (", getdual(Ball1)*2*getsol(x), ",",getdual(Ball1)*2*getsol(y),")") writeln("Linearization of the objective at the optimal solution: (0,1)") writeln("-----------------------------------------------") writeln ! Now introduce a new nonlinear constraint that only intersects the first ! constraint in a single point Ball2 := (x-2)^2 + y^2 <= 1 minimize(y) writeln("Optimal solution: x=",getsol(x)," y=",getsol(y)) ! Look at the dual values, and note the very large numbers ! Note that the solver kept pertubing y even though the very strict ! complementarity target writeln("Optimal solution: pi(Ball1)=",getdual(Ball1), " pi(Ball2)=",getdual(Ball2)) writeln("-----------------------------------------------") writeln ! The solver has artifically pertuebed (relaxed) the problem. ! Let we do the relaxation instead by the means of redefining the second constraints Ball2 := (x-2)^2 + y^2 <= 1.001 minimize(y) writeln("Optimal solution: x=",getsol(x)," y=",getsol(y)) ! Even though the relaxation is small, 0.1%, the duals are now meaningful writeln("Optimal solution: pi(Ball1)=",getdual(Ball1), " pi(Ball2)=",getdual(Ball2)) writeln("-----------------------------------------------") writeln ! The case of x^2 <= 0 is interseting, as there is simply no constraint gradient ! The effect is the same: the solver perturbs sethidden(Ball1, true) sethidden(Ball2, true) Ball := x^2 + y^2 <= 0 minimize(y) writeln("Optimal solution: x=",getsol(x)," y=",getsol(y)) ! Even though the relaxation is small, 0.1%, the duals are now meaningful writeln("Optimal solution: pi(Ball)=",getdual(Ball)) writeln("-----------------------------------------------") writeln end-model | |||||||||

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