In this paper, we provide some new necessary and sufficient conditions for pseudoconvexity and semistrict quasiconvexity of a given proper extended real-valued function in terms of the Clarke-Rockafellar subdifferenti...
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In this paper, we provide some new necessary and sufficient conditions for pseudoconvexity and semistrict quasiconvexity of a given proper extended real-valued function in terms of the Clarke-Rockafellar subdifferential. Further, we extend to programs with pseudoconvex objective function two earlier characterizations of the solutions set of a set constrained nonlinear programming problem due to Mangasarian (Oper Res Lett 7:21-26, 1988). A positive function p appears in the most results. It is replaced by the number 1 if the function is convex and its domain of definition is convex, too.
This paper is devoted to constructing Wolfe and Mond-Weir dual models for interval-valued pseudoconvex optimization problem with equilibrium constraints, as well as providing weak and strong duality theorems for the s...
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This paper is devoted to constructing Wolfe and Mond-Weir dual models for interval-valued pseudoconvex optimization problem with equilibrium constraints, as well as providing weak and strong duality theorems for the same using the notion of contingent epiderivatives with pseudoconvex functions in real Banach spaces. First, we introduce the Mangasarian-Fromovitz type regularity condition and the two Wolfe and Mond-Weir dual models to such problem. Second, under suitable assumptions on the pseudoconvexity of objective and constraint functions, weak and strong duality theorems for the interval-valued pseudoconvex optimization problem with equilibrium constraints and its Mond-Weir and Wolfe dual problems are derived. An application of the obtained results for the GA-stationary vector to such interval-valued pseudoconvex optimization problem on sufficient optimality is presented. We also give several examples that illustrate our results in the paper.
A class of functions called pseudo B-vex and quasi B-vex functions is introduced by relaxing the definitions of B-vex, pseudoconvex, and quasiconvex functions. Similarly, the class of B-invex, pseudo B-invex, and quas...
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A class of functions called pseudo B-vex and quasi B-vex functions is introduced by relaxing the definitions of B-vex, pseudoconvex, and quasiconvex functions. Similarly, the class of B-invex, pseudo B-invex, and quasi B-invex functions is defined as a generalization of B-vex, pseudo B-vex, and quasi B-vex functions. The sufficient optimality conditions and duality results are obtained for a nonlinear programming problem involving B-vex and B-invex functions.
In Ref. 1, Bazaraa and Goode provided an algorithm for solving a nonlinear programming problem with linear constraints. In this paper, we show that this algorithm possesses good convergence properties.
In Ref. 1, Bazaraa and Goode provided an algorithm for solving a nonlinear programming problem with linear constraints. In this paper, we show that this algorithm possesses good convergence properties.
We establish connections between some concepts of generalized monotonicity for set-valued maps introduced earlier and some notions of generalized convexity. Moreover, a notion of pseudomonotonicity for set-valued maps...
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We establish connections between some concepts of generalized monotonicity for set-valued maps introduced earlier and some notions of generalized convexity. Moreover, a notion of pseudomonotonicity for set-valued maps is introduced;it is shown that, if a function f is continuous, then its pseudoconvexity is equivalent to the pseudomonotonicity of its generalized subdifferential in the sense of Clarke and Rockafellar.
We study the convergence properties of a (block) coordinate descent method applied to minimize a nondifferentiable (nonconvex) function f(x(I),..., x(n)) with certain separability and regularity properties. Assuming t...
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We study the convergence properties of a (block) coordinate descent method applied to minimize a nondifferentiable (nonconvex) function f(x(I),..., x(n)) with certain separability and regularity properties. Assuming that f is continuous on a compact level set, the subsequence convergence of the iterates to a stationary point is shown when either f is pseudoconvex in every pair of coordinate blocks from among N - 1 coordinate blocks orf has at most one minimum in each of N - 2 coordinate blocks. If f is quasiconvex and hemivariate in every coordinate block, then the assumptions of continuity off and compactness of the level set may be relaxed further. These results are applied to derive new land old) convergence results for the proximal minimization algorithm, an algorithm of Arimoto and Blahut, and an algorithm of Han. They are applied also to a problem of blind source separation.
For solving a broad class of nonconvex programming problems on an unbounded constraint set, we provide a self-adaptive step-size strategy that does not include line-search techniques and establishes the convergence of...
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For solving a broad class of nonconvex programming problems on an unbounded constraint set, we provide a self-adaptive step-size strategy that does not include line-search techniques and establishes the convergence of a generic approach under mild assumptions. Specifically, the objective function may not satisfy the convexity condition. Unlike descent line-search algorithms, it does not need a known Lipschitz constant to figure out how big the first step should be. The crucial feature of this process is the steady reduction of the step size until a certain condition is fulfilled. In particular, it can provide a new gradient projection approach to optimization problems with an unbounded constrained set. To demonstrate the effectiveness of the proposed technique for large-scale problems, we apply it to some experiments on machine learning, such as supervised feature selection, multi-variable logistic regressions and neural networks for classification.
The cutting-plane optimization methods rely on the idea that any subgradient of the objective function or the active/violated constraints defines a halfspace to be excluded from a set that contains an optimal solution...
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The cutting-plane optimization methods rely on the idea that any subgradient of the objective function or the active/violated constraints defines a halfspace to be excluded from a set that contains an optimal solution: the localizing set. This algorithm converges towards a global minimum of any pseudoconvex subdifferentiable function. A naive extension for multiobjective optimization would be using simultaneously some subgradients of all objective functions for a given feasible point. However, as demonstrated in this paper, this approach can lead to a convergence towards non-optimal points. This paper introduces an optimization strategy for cutting-plane methods to cope with multiobjective problems without any scalarization procedure. The proposed strategy guarantees that its optimal solution is a Pareto Optimal solution of the original problem, which is also no worse than the starting point, and that any Pareto Optimal solution can be sampled. Moreover, the auxiliary problem is infeasible only if the original problem is also infeasible. The new strategy inherits the original theoretical guarantees of cutting planes methods and it can be applied to build other strategies. (C) 2019 Elsevier B.V. All rights reserved.
Some equivalent conditions for convexity of the solution set of a pseudoconvex inequality are presented. These conditions turn out to be very useful in characterizing the solution sets of optimization problems of pseu...
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Some equivalent conditions for convexity of the solution set of a pseudoconvex inequality are presented. These conditions turn out to be very useful in characterizing the solution sets of optimization problems of pseudoconvex functions defined on Riemannian manifold.
The purpose of this paper is to introduce a new class of generalized cone-pseudoconvex functions and strongly cone-pseudoconvex functions, called second order (K,F)-pseudoconvex functions and strongly second order (K,...
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The purpose of this paper is to introduce a new class of generalized cone-pseudoconvex functions and strongly cone-pseudoconvex functions, called second order (K,F)-pseudoconvex functions and strongly second order (K,F)-pseudoconvex functions. A pair of second order symmetric dual multiobjective nonlinear programs is formulated over arbitrary generalized cone-pseudoconvex functions. For these second order symmetric dual programs, the weak, strong and converse duality theorems are established using the above generalization of cone-pseudoconvex functions. A self duality theorem is also given by assuming the functions involved to be skew-symmetric. (C) 2012 Elsevier Inc. All rights reserved.
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