PSET 4 SOL - MS&E 111 Introduction to Optimization...

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MS&E 111: Introduction to OptimizationInstructor: Professor Amin Saberi ([email protected])HW#4 – Due Thursday, May 8thHomework drop-o:you should deposit your homework in MS&E 111 drop-obox locatedin Huang basement before 12:30 pm on the due date.We do not collect any homework in class.1. Read VRM, Section 3.4.(a) Convert the following optimization problemmax-x1-2x2-x3s.t.x1+x2+x3= 1|x1|4x1, x2, x30into a linear program of the formmaxcTxs.t.Axbx0Note:|x|denotes the absolute value ofx.(b) Find the dual of this LP.Solution:
(b) The dual is by definitionminbTys.t.ATycy0and can be written in a developed formminy1-y2+ 4y3s.t.y1-y2+y3≥ -1y1-y2≥ -2y1-y2≥ -1y1, y2, y302. Find the dual ofminuTzs.t.Bz=vzawherezis the vector of decision variables.u,v,aandBare constants.[Hint: Look at VRM Section 4.3 to define the dual of general LPs]
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