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Let X be a locally convex space, f:X convex function, and G c. Proof The proof is similar to that of Corollary 3. Let X be a locally convex space, f:X coajfine function, and G c.

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The proof of the second equality of 3. Note that in contrast to 3. Duality for Quasi-convex Supremization Hence, by Theorem 3. Let X be a locally convex space, f an evenly quasi-convex function satisfying 1. Then, by Theorem 3. Indeed, let us give here an example of such a result. Hence, formula 3.

## Duality for Nonconvex Approximation and Optimization

Duality for Quasi-convex Supremization whence, by Remark 3. Indeed, by 3. Thus, problem 3. We shall define a dual problem to the primal supremization problem P of 3. For the constrained primal supremization problem 3. Now we shall study some more structured ways of expressing the primal constraint sets G c X. In the present section we shall consider one of these ways, namely that of systems, and surrogate and Lagrangian duality for supremization in systems.

Therefore, in the sequel we shall assume, without any special mention, that w X nr 0. Duality for Quasi-convex Supremization b Problem 3. Indeed, given a system X, Z, u and T, f as above, problem 3. However, in the study of the "mathematical programming problem" 3. There are several natural ways to introduce unconstrained dual problems to 3. In the converse direction, given any X, Z, u and T as above, by 3. If we have inf f x inf fix , 3. Duality for Quasi-convex Supremization with possibly strict inequality. Therefore, by 3. Hence, applying Theorems 3.

Concerning Lagrangian duality for the primal supremization problem 3. From 1. By Remark 4. For an element go e G, the following statements are equivalent: r. Condition 4. D Now we shall give some subdifferential characterizations of maximum points. To this end, let us first introduce the following class of abstract quasi-convex functions: Definition 4. Remark 4.

## Duality for Nonconvex Approximation and Optimization | Mathematical Association of America

Proposition 4. Optimal Solutions for Quasi-convex Maximization Theorem 4. D Proposition 4. By the above proof of Proposition 4.

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But by 4. Let X be a locally convex space, f an upper semicontinuous quasiconvex function satisfying 1. D Remark 4. Note that in this case Proposition 4. Hence by 4. Let us denote XQ of Theorem 2. Optimal Solutions for Quasi-convex Maximization 4. The assumption 1. Conversely, every hyperplane that supports the set S is of the form 4.

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In this case, Corollary 4. Proof If xo satisfies 4.

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For an element go e G the following statements are equivalent: P. Furthermore, by Remark 4. Optimal Solutions for Quasi-convex Maximization Remark 4. Theorem 4. The optimal functions are nothing other than the optimal solutions of the dual problem 3. Corollary 4. D Corollary 4. As shown by Example 2. Part a follows from the necessity part of Corollary 4.

Part b is shown by Remark 4. Part c is Corollary 4. We recall that a subset G of a normed linear space X is called proximinal if each XQ e X admits a best approximation in G, i. Therefore, Oo c — x 4. Thus, 4. D Theorem 4.

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Hence by Theorem 1. Consequently, 4. D Replacing in 4. Namely, we have the following theorem: Theorem 4. Indeed, we have the following: Theorem 4.

http://co.organiccrap.com/90110.php For an element go G G, the following statements are equivalent: 4. Optimal Solutions for Quasi-convex Maximization r. On the other hand, by the first inequahty in 4. Namely, it is known see the Notes and Remarks to Section 5.