The problem data are x f and g explain why the optimal

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Unformatted text preview: problem of (17) and express it in terms of the Lagrange dual function g for problem (14). 35 (c) Use the result in (b) to prove the following property. If t > 1T λ⋆ , then any minimizer of (16) is also an optimal solution of (14). (The second term in (16) is called a penalty function for the constraints in (14). It is zero if x is feasible, and adds a penalty to the cost function when x is infeasible. The penalty function is called exact because for t large enough, the solution of the unconstrained problem (16) is also a solution of (14).) 4.19 Infimal convolution. Let f1 , . . . , fm be convex functions on Rn . Their infimal convolution, denoted g = f1 ⋄ · · · ⋄ fm (several other notations are also used), is defined as g (x) = inf {f1 (x1 ) + · · · + fm (xm ) | x1 + · · · + xm = x}, with the natural domain (i.e., defined by g (x) < ∞). In one simple interpretation, fi (xi ) is the cost for the ith firm to produce a mix of products given by xi ; g (x) is then the optimal cost obtained if the firms can...
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