Topic 19 - Intro to Lagrangian Duality

214 note that we could have formulated and solved the

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Unformatted text preview: } (0,0) -3/2 0 (0,1) -1/2 -2 (1,0) -1/2 -1 (1,1) 1/2 -3 Therefore, we have { seeks the maximum value of }. The outer objective subject to . 215 We have seen that a duality gap can occur for integer programming problems. We can use the geometrical approach to show that a duality gap can exist for nonlinear programming problems as well. This is illustrated in Figure 6.3 of the BSS textbook. z X x G [ g(x), f (x)] Duality gap Optimal primal objective Optimal dual objective v (y ) y Figure 6.3 (BSS textbook, page 264) Relationship Between Objective Values for the Primal and Lagrangian Dual Problems For linear programming, if both the primal and dual problems are feasible, then they have equal optimal objective values. Unfortunately, as we have just seen, that is not always the case in nonlinear programming problems. Recall the problems we are addressing. P: minimize subject to D: maximize subject to where for , { ∑ } { Like LP, however, the less restrictive condition of weak duality still holds. 216 } Theorem 6.2.1 (Weak Duality): If ̅ is feasible to P and ̅ is feasible to D, then ̅ ̅. Proof: Corollary 1: ̅ { ̅ } ̅ ̅ ̅ ̅ ̅ is an upper bound for the optimal objective value of D, and lower bound for the optimal objective value of P. ̅ is a Interpretation: Any feasible solution to one problem gives a best-case bound on the other. Corollary 2: If ̅ is feasible to P, ̅ is feasible to D, and ̅ solves D. ̅ ̅ , then ̅ solves P and Interpretation: If P, D are feasible with equal objective values, then both are optimal. Corollary 3: If the objective of D is unbounded, then P is infeasible. Interpretation: Otherwise, if P is feasible, then it would provide an upper bound on D, which would contradict that D is unbounded. Corollary 4: If the objective of P is unbounded, then for all . Interpretation: D is never infeasible, since is always possible. If P’s objective value is unbounded ( ) and we know that , then we must have . Duality Gap: As mentioned previously, Problems P and D need not have the same optimal objective value. In this case, the problems are said to have a duality gap, which equals the difference in their optimal objective values. 217 In some cases, however, the problems can be shown to have no duality gap. Strong Duality Theorem (Theorem 6.2.4): If f and g are convex functions, X is a convex set, and Slater’s CQ holds (there is a strict interior point for the inequality constraints), then there is no duality gap, meaning that P and D have equal optimal objective values. Implication: For convex programs with non-empty interiors, there is no duality gap. Note: This theorem is proved in the textbook. It is a fairly straightforward proof, but we omit it due to time considerations. Another condition can be examined to d...
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