Interpretation and extensions. Duality Theory

5.2. DUALITY THEORY 63 M ˆb − ˆλ ′ ˆb ≥ f x − ˆλ ′ f x or M ˆb ≥ f x − ˆλ ′ f x − ˆb , so that M ˆb ≥ sup{f x − ˆλ ′ f x − ˆb|x ∈ X} = mˆλ . By weak duality Lemma 2 it follows that M ˆb = mˆ λ and ˆ λ is optimal for 5.19. Conversely suppose ˆ λ ≥ 0, and mˆλ = Mˆb. Then for any x ∈ X M ˆb ≥ f x − ˆλ ′ f x − ˆb , and if moreover f x ≤ b, then ˆλ ′ f x − b ≤ 0, so that M ˆb ≥ f x − ˆλ ′ f x − ˆb + ˆλ ′ f x − b = f x − ˆλ ′ b + ˆ λ ′ ˆb for x ∈ Ωb . Hence, M b = sup{f x|x ∈ Ωb} ≤ Mˆb + ˆλ ′ b − ˆb , so that ˆ λ is a supergradient. ♦ We can now summarize our results as follows. Theorem 2: Duality Suppose ˆ b ∈ B, Mˆb ∞, and M is stable at ˆb. Then i there exists an optimal solution ˆ λ for the dual, and mˆ λ = M ˆb, ii ˆ λ is optimal for the dual iff ˆ λ is a supergradient of M at ˆb, iii if ˆ λ is any optimal solution for the dual, then ˆ x is optimal for the primal iff ˆ x, ˆ λ satisfy the optimality conditions of 5.21, 5.22, and 5.23. Proof: i follows from Lemmas 4,6. ii is implied by Lemma 6. The “if” part of iii follows from Theorem 1, whereas the “only if” part of iii follows from Lemma 5. ♦ Corollary 1: Under the hypothesis of Theorem 2, if ˆ λ is an optimal solution to the dual then ∂M + ∂b i ˆb ≤ ˆλ i ≤ ∂M − ∂b i ˆb. Exercise 4: Prove Corollary 1. Hint: See Theorem 5 of 4.2.3.

5.2.2 Interpretation and extensions.

It is easy to see using convexity properties that, if X = R n and f i , 0 ≤ i ≤ m, are differentiable, then the optimality conditions 5.21, 5.22, and 5.23 are equivalent to the Kuhn-Tucker condition 5.8. Thus the condition of stability of M at ˆb plays a similar role to the constraint qualification. However, by Lemmas 4, 6 stability is equivalent to the existence of optimal dual variables, whereas CQ is only a sufficient condition. In other words if CQ holds at ˆ x then M is stable at ˆb. In particular, if X = R n and the f i are differentiable, the various conditions of Section 1.3 imply stability. Here we give one sufficient condition which implies stability for the general case. Lemma 7: If ˆ b is in the interior of B, in particular if there exists x ∈ X such that f i x ˆb i for 1 ≤ i ≤ m, then M is stable at ˆb. 64 CHAPTER 5. NONLINEAR PROGRAMMING The proof rests on the Separation Theorem for convex sets, and only depends on the fact that M is concave, M ˆb ∞ without loss of generality, and ˆb is the interior of B. For details see the Appendix. Much of duality theory can be given an economic interpretation similar to that in Section 4.4. Thus, we can think of x as the vector of n activity levels, f x the corresponding revenue, X as constraints due to physical or long-term limitations, b as the vector of current resource supplies, and finally f x the amount of these resources used up at activity levels x. The various convexity conditions are generalizations of the economic hypothesis of non-increasing returns-to-scale. The primal problem 5.17 is the short-term decision problem faced by the firm. Next, if the current resources can be bought or sold at prices ˆ λ = λ 1 , . . . , λ m ′ , the firm faces the decision problem 5.18. If for a price system ˆ λ, an optimal solution of 5.17 also is an optimal solution for 5.18, then we can interpret ˆ λ as a system of equilibrium prices just as in 4.2. Assuming the realistic condition ˆ b ∈ B, Mˆb ∞ we can see from Theorem 2 and its Corollary 1 that there exists an equilibrium price system iff ∂M + ∂b i ˆb ∞, 1 ≤ i ≤ m; if we interpret ∂M + ∂b i ˆb as the marginal revenue of the ith resource, we can say that equilibrium prices exist iff marginal productivities of every variable resource is finite. These ideas are developed in Gale [1967]. . M b A b ˆb M ˆb Figure 5.5: If M is not concave there may be no supporting hyperplane at M ˆb, ˆb. Referring to Figure 5.3 or Figure 5.4, and comparing with Figure 5.5 it is evident that if M is not concave or, equivalently, if its hypograph A is not convex, there may be no hyperplane supporting A at M ˆb, ˆb. This is the reason why duality theory requires the often restrictive convexity hypoth- esis on X and f i . It is possible to obtain the duality theorem under conditions slightly weaker than convexity but since these conditions are not easily verifiable we do not pursue this direction any fur- ther see Luenberger [1968]. A much more promising development has recently taken place. The basic idea involved is to consider supporting A at M ˆb, ˆb by non-vertical surfaces ˆ π more gen- eral than hyperplanes; see Figure 5.6. Instead of 5.18 we would then have more general problem of the form 5.26: Maximize f x − F f x − ˆb subject to x ∈ X , 5.26 5.2. DUALITY THEORY 65 where F : R m → R is chosen so that ˆπ in Figure 5.6 is the graph of the function b 7→ Mˆb − F b − ˆb. Usually F is chosen from a class of functions φ parameterized by µ = µ 1 , . . . , µ k ≥ 0. Then for each fixed µ ≥ 0 we have 5.27 instead of 5.26: Maximize f x − φµ; f x − ˆb subject to x ∈ X . 5.27 . M b ˆ π A b ˆb M ˆb Figure 5.6: The surface ˆ π supports A at M ˆb, ˆb. If we let ψµ =sup{f x − φµ; f x − ˆb|x ∈ X} . then the dual problem is Minimize ψµ subject to µ ≥ 0 , in analogy with 5.19. The economic interpretation of 5.27 would be that if the prevailing non-uniform price system is φµ; · then the resources f x − ˆb can be bought or sold for the amount φµ; fx − ˆb. For such an interpretation to make sense we should have φµ; b ≥ 0 for b ≥ 0, and φµ; b ≥ φµ; ˜b whenever b ≥ ˜b. A relatively unnoticed, but quite interesting development along these lines is presented in Frank [1969]. Also see Arrow and Hurwicz [1960]. For non-economic applications, of course, no such limitation on φ is necessary. The following references are pertinent: Gould [1969], Greenberg and Pierskalla [1970], Banerjee [1971]. For more details concerning the topics of 2.1 see Geoffrion [1970a] and for a mathematically more elegant treatment see Rockafellar [1970].

5.2.3 Applications.