In this paper we introduce an algorithm for solving variational inequality problems when the operator is pseudomonotone and point-to-set (therefore not relying on continuity assumptions). Our motivation is the develop...
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In this paper we introduce an algorithm for solving variational inequality problems when the operator is pseudomonotone and point-to-set (therefore not relying on continuity assumptions). Our motivation is the development of a method for solving optimisation problems appearing in Chebyshev rational and generalised rational approximation problems, where the approximations are constructed as ratios of linear forms (linear combinations of basis functions). The coefficients of the linear forms are subject to optimisation and the basis functions are continuous functions. It is known that the objective functions in generalised rational approximation problems are quasiconvex. In this paper we prove a stronger result, the objective functions are pseudoconvex in the sense of Penot and Quang. Then we develop numerical methods, that are efficient for a wide range of pseudoconvex functions and test them on generalised rational approximation problems.
The vector optimization problem may have a nonsmooth objective function. Therefore, we introduce the Minty vector variational inequality (Minty VVI) and the Stampacchia vector variational inequality (Stampacchia VVI) ...
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The vector optimization problem may have a nonsmooth objective function. Therefore, we introduce the Minty vector variational inequality (Minty VVI) and the Stampacchia vector variational inequality (Stampacchia VVI) defined by means of upper Dini derivative. By using the Minty VVI, we provide a necessary and sufficient condition for a vector minimal point (v.m.p.) of a vector optimization problem for pseudoconvex functions involving Dini derivatives. We establish the relationship between the Minty VVI and the Stampacchia VVI under upper sign continuity. Some relationships among v.m.p., weak v.m.p., solutions of the Stampacchia VVI and solutions of the Minty VVI are discussed. We present also an existence result for the solutions of the weak Minty VVI and the weak Stampacchia VVI.
We investigate the variational inequality with pseudomonotone operators (in the sense of Karamardian) in Banach spaces. New existence results which extend many known results in infinite-dimensional spaces are derived ...
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We investigate the variational inequality with pseudomonotone operators (in the sense of Karamardian) in Banach spaces. New existence results which extend many known results in infinite-dimensional spaces are derived under rather weak assumptions. New uniqueness results which also seem to be new even in finite-dimensional spaces are also derived. In particular, new existence and uniqueness results for the complementarity problem with pseudomonotone operators in Banach spaces are obtained. Also, existence and uniqueness results for minimization problems with pseudoconvex functions in Banach spaces are obtained.
The article proposes an exact approach to finding the global solution of a nonconvex semivectorial bilevel optimization problem, where the objective functions at each level are pseudoconvex, and the constraints are qu...
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The article proposes an exact approach to finding the global solution of a nonconvex semivectorial bilevel optimization problem, where the objective functions at each level are pseudoconvex, and the constraints are quasiconvex. Due to its non-convexity, this problem is challenging, but it attracts more and more interest because of its practical applications. The algorithm is developed based on monotonic optimization combined with a recent neurodynamic approach, where the solution set of the lower-level problem is inner approximated by copolyblocks in outcome space. From that, the upper-level problem is solved using the branch-and-bound method. Finding the bounds is converted to pseudoconvex programming problems, which are solved using the neurodynamic method. The algorithm's convergence is proved, and computational experiments are implemented to demonstrate the accuracy of the proposed approach.
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.
In this paper, we give some properties for nondifferentiable pseudoconvex functions on Hadamard manifolds, and discuss the connections between pseudoconvex functions and pseudomonotone vector fields. Moreover, we stud...
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In this paper, we give some properties for nondifferentiable pseudoconvex functions on Hadamard manifolds, and discuss the connections between pseudoconvex functions and pseudomonotone vector fields. Moreover, we study Minty and Stampacchia vector variational inequalities, which are formulated in terms of Clarke subdifferential for nonsmooth functions. Some relations between the vector variational inequalities and nonsmooth vector optimization problems are established under pseudoconvexity or pseudomonotonicity. The results presented in this paper extend some corresponding known results given in the literatures.
In this article, two types of cone-pseudomonotone bifunctions have been introduced and the weaker form of pseudomonotonicity is used to establish an existence theorem for a Stampacchia-kind vector variational inequali...
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In this article, two types of cone-pseudomonotone bifunctions have been introduced and the weaker form of pseudomonotonicity is used to establish an existence theorem for a Stampacchia-kind vector variational inequality problem given in terms of bifunctions. For a vector optimization problem, the necessary and sufficient optimality conditions in terms of an associated vector variational inequality problem have been established using a generalized form of cone pseudoconvexity of objective function.
pseudoconvexity and strict pseudoconvexity concepts are extended to nondifferentiable non-locally Lipschitz setting in ℝn by means of Clarke's generalized gradients and Rockafellar's asymptotic gradients. New ...
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