Question

Problem 3 (Convex Optimization): Consider a linear programming: min c'e s.t.Ax > b (1) x > 0 Find the dual problem of the linear programming and argue that: (1) If the primal is unbounded, then the dual is infeasible; (2) If the primal is infeasible, then the dual is either infeasible or unbounded. 1 Note that strong duality holds for a linear programming if either the primal or the dual is feasible. In other words, for a linear programming, the only exception to strong duality occurs when p* = o or d* = -00.

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Answer 1

consider linear program min eta 2 D At An>b mo the duce of boty maa y st: 0) C Then Aty = 0 y so the primal is unbounded, let if the duais We'll prove the result het frauible, and bounded by contradiction Assume Aty = has a solution y. by weak duality, bly is a lower bound for all solutions to the primal je cashy but it is the fact that puimial is un boundled. so, o use assumption that is feasible was wrong. Hence dual is infeasible. the primal he infrasible. to show that dual e either infeasible or unbounded the contradiction to dual (2) W cs Scanned with CamScanner

Let dual is if dual is if and feasible bounded . By using Strong duality theorem; feasible and bounded , so is primal, so again we get contradiction Henu, if primal is unfeasible then dual is either unbounded ou infeasible. a CS Scanned with CamScanner

Weak theorem states:- The value for any feasible solution is always greater function of objective of mihimization primae problem that equal to of its dual. il, Let primal problem be min cta st Auzb a>, o the respected dual problem is 677 man St. Aly to 430 by theacem, So, c'xsbty Strong duality theorem: 9. premcl has a finite optimal suultion then dual also has a finite optimal solution with same objectwe function value 3) g pumal and dual both have feasible solutions, there pumal has finite solutibu coptimal)x* and dual has finite optimal solution it and his the optimal values of pulmal and dual al equal. CS Scanned with CamScanner i)