1,721,016 research outputs found
Componentwise versus Global Aproaches to Nonsmooth Multiobjective Optimization
No full textThe paper aims to analyse and compare two different approaches to optimality conditions for multiobjective optimization problems, which involve nonsmooth functions. Both necessary and sufficient first order conditions are presented for the case in which constraint is given just as a set. Finally, the optimality conditions based on these two approaches are compared
On Sufficient Second Order Optimality Conditions in Multiobjective Optimization
No full textA second order sufficient optimality criterion is presented for a multiobjective problem subject to a constraint given just as a set. To this aim, we first refine known necessary conditions in such a way that the sufficient ones differ by the replacement of inequalities by strict inequalities. Furthermore, we show that no relationship holds between this criterion and a sufficient multipliers rule, when the constraint is described by inequalities and equalities. Finally, improvements of this criterion for the unconstrained case are presented, stressing the differences with single-objective optimizatio
D-gap functions and descent techniques for solving quilibrium problems
No full textA new algorithm for solving equilibrium problems with differentiable bifunctions is provided. The algorithm is based on descent directions of a suitable family of D-gap functions. Its convergence is proved under assumptions which do not guarantee the equivalence between the stationary points of the D-gap functions and the solutions of the equilibrium problem. Moreover, the algorithm does not require to set parameters according to thresholds which depend on regularity properties of the equilibrium bifunction. Finally, the results of preliminary numerical tests on Nash equilibrium problems with quadratic payoffs are reported
Gap functions and penalization for solving equilibrium problems with nonlinear constraints
No full textThe paper deals with equilibrium problems (EPs) with nonlinear convex constraints. First, EP is reformulated as a global optimization problem introducing a class of gap functions, in which the feasible set of EP is replaced by a polyhedral approximation. Then, an algorithm is given for solving EP through a descent type procedure related to exact penalties of the gap functions and its global convergence is proved. Finally, the algorithm is tested on a network oligopoly problem with nonlinear congestion constraints<br /
Gap functions for quasi-equilibria
Full text linkAn approach for solving quasi-equilibrium problems (QEPs) is proposed relying on gap functions, which allow reformulating QEPs as global optimization problems. The (generalized) smoothness properties of a gap function are analysed and an upper estimate of its Clarke directional derivative is given. Monotonicity assumptions on both the equilibrium and constraining bifunctions are a key tool to guarantee that all the stationary points of a gap function actually solve QEP. A few classes of constraints satisfying such assumptions are identified covering a wide range of situations. Relying on these results, a descent method for solving QEP is devised and its convergence proved. Finally, error bounds are given in order to guarantee the boundedness of the sequence generated by the algorithm
A close look at auxiliary problem principles for equilibria
No full textThe auxiliary problem principle allows solving a given equilibrium problem (EP) through an equivalent auxiliary problem with better properties. The paper investigates two families of auxiliary EPs: the classical auxiliary problems, in which a regularizing term is added to the equilibrium bifunction, and the regularized Minty EPs. The conditions that ensure the equivalence of a given EP with each of these auxiliary problems are investigated. This analysis leads to extending some known results for variational inequalities and linear EPs to the general case; moreover, new results are obtained as well. In particular, both new results on the existence and uniqueness of solutions and new error bounds based on gap functions with good convexity properties are obtained under weak quasimonotonicity or weak concavity assumptions
Regularity conditions for the linear separation of sets
No full textIn this paper, we study the linear separation between a set and a convex cone. We introduce the concepts of regularity and total regularity of the separation with respect to a face of the cone and we give theorems characterizing them
A successive linear programming algorithm for nonsmooth monotone variational inequalities
No full textAn algorithm for solving nonsmooth monotone variational inequalities subject to linear constraints is proposed. Combining a cutting plane procedure for strictly monotone variational inequalities with the Tikhonov regularization technique, we devise an algorithm based on successive linear programming. Preliminary numerical results are reported
Regularity Conditions in Vector Optimization
No full textBy exploiting very recent results concerning linear separation between a set and a convex cone, the image space approach allows us to achieve new regularity conditions for vector minimum problems. In particular, we obtain very general Mangasarian-Fromovitz type
conditions
Differentiated oligopolistic markets with concave cost functions via Ky Fan inequalities
Full text linkA model for Nash-Cournot oligopolistic markets with concave cost functions and a differentiated commodity is analysed. Equilibrium states are characterized through Ky Fan inequalities. Relying on the minimization of a suitable merit function, a general algorithmic scheme for solving them is provided. Two concrete algorithms are therefore designed that converge under suitable convexity and monotonicity assumptions. The results of preliminary numerical tests on randomly generated markets are also reported
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