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- Topics in Nonconvex Optimization: Theory and Applications | Mathematical Association of America.
- Topics in Nonconvex Optimization : Shashi Kant Mishra : ?
- 6th World Congress on Global Optimization (WCGO ) - midhighdothank.tk.
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- Machine Learning and Optimization (Redmond).
See also Sect. We report only the following one.
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Let x0 be feasible for P and the Kuhn—Tucker conditions 5. Weak duality between P and D1 thus requires a stronger assumption than weak duality between P and D2. Then Proof. These authors, under various C-invexity assumptions, obtained weak and strong duality results for P and D3. The saddle point problem, when the functions involved are nonsmooth and C-invex, has been treated by Jeyakumar  and by Brandao et al.
Mathematical Theory of Optimization
Brandao et al. Following Brandao et al. The left side inequality of 5. The following convention for equalities and inequalities will be used.
MATHEMATICAL OPTIMIZATION THEORY AND OPERATIONS RESEARCH
The study of the solutions of a multiobjective problem may be approached from two aspects: a Trying to relate them with the solutions to he scalar problems, whose resolution has been studied extensively see, e. As much in one case as in the other, the convexity plays an important role as a fundamental condition in obtaining the desired results. Second and higher order duality results are also discussed for multiobjective case. Moreover, we discuss multi- objective symmetric duality involving invex and generalized invex functions.
In multiobjective vector optimization problems, multiobjectives are usu- ally noncommensurable and cannot be combined into a single objective. Osuna-Gomez et al. Note that stationary points for a scalar optimization problem are those whose vector gradients are zero.
However, vector critical points are those such that there exists a non-negative linear combination of the gradient vectors of each component objective function, valued at that point, equal to zero. Theorem 6. For the converse of the above theorem, we need some convexity hypothesis. See Osuna-Gomez et al. For the necessary part see Osuna-Gomez et al.
See the proof of Theorem 2. Martin  established that it is not possible to infer f and g -invexity from the equivalence of minimum points and Kuhn—Tucker points for scalar programming problems.
Similar question was investigated by Osuna-Gomez et al. KT-invex functions are extended to multiob- jective case by Osuna-Gomez et al. An analogous theorem for vector Kuhn—Tucker points is as follows see,  : Theorem 6. Even a stronger result is given by Osuna-Gomez et al.
For the proof, see  or . Several results from scalar optimization problems can be extended to vector optimization problem. Let K denote the set of feasible solutions of the above vector maximum problem. Singh and Hanson  extended the concept to situation where M depends on x.
For other algebraic properties of pre-invex functions, see Sect. Later Weir  established the following result: Theorem 6. Remark 6. They have also obtained duality results for Wolfe type and Mond—Weir type duals under the aforesaid weaker invexity assump- tions. We give some results from Kaul et al. See Kaul et al. They have also given the following necessary optimality conditions. This requirement turns out to be a severe restriction in applications.https://tageridingvire.tk
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Because of this restriction, pseudo-linear multiobjective problems  and certain nonlinear multiobjective fractional programming problems require separate treatment as far as optimality and duality properties are concerned. In order to avoid this restriction, Jeyakumar and Mond  introduced a new class of functions, namely V -invex functions. For detailed study on V -invex functions and applications see, Mishra . The V -invex functions can be formed from certain nonconvex functions in particular from convex-concave or linear fractional functions by coordinate transformations, see, e.
Following Kaul et al. If strict inequality holds in 6. Hanson et al.
However, these authors have only discussed duality results under these new classes of functions. Example 6. Let us give the following second order optimality conditions for multiobjective optimization problem vector minimization problem : Theorem 6. For further discussion on univex functions see also Rueda et al. The following Wolfe type dual model is an extension of the Wolfe dual problem see Sect. Weir  proved that weak duality and strong duality hold for VP and WD. Weir and Mond  obtained the above two duality relation under pre- invexity assumptions as well.
Following Weir  Kaul et al. The Mond—Weir type dual for VP has the advantage over the Wolfe dual is that it allows further weakening of the invexity conditions in order to obtain usual duality results. However, the converse is not necessarily true, as can be seen from the following example from . Now we are in position to state the weak duality given by Aghezzaf and Hachimi .
Suppose contrary to the result, i. By feasibility and duality constraint 6. We refer the reader to Aghezzaf and Hachimi . We give the following strong and converse duality theorems from Aghezzaf and Hachimi  under the invexity assumptions used in the above weak duality theorem. If also the invexity hypothesis used in the weak duality Theorem 6.
Aghezzaf and Hachimi  have extended the results of Aghezzaf and Hachimi  to generalized type I invex- ity assumptions. Let us recall the following concept of weaker type I invexity from . Aghezzaf and Hachimi  have shown that every strictly-pseudo-quasi-type I functions are weak strictly-pseudo-quasi-type I but the converse is not neces- sarily true.
See, e. These authors have obtained several weak, strong and converse duality for VP and MWD under various weaker type I invexity assumptions. Aghezzaf and Hachimi  have obtained several other weak duality results under other weaker invexity assumptions. Based on the Theorem 6.
If the generalized invexity hypothesis of weak duality Theorem 6. They have extended the concept of pseudo-quasi-type I functions to pseudo-quasi-V -type I by combining the con- cept of pseudo-quasi-type I functions and V -invex functions. For proof, see Hanson et al.
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