We study the relationships between three concepts which arise in connection with variational inequality problems: central paths defined by arbitrary barriers, generalized proximal point methods (where a Bregman distan...
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We study the relationships between three concepts which arise in connection with variational inequality problems: central paths defined by arbitrary barriers, generalized proximal point methods (where a Bregman distance substitutes for the Euclidean one), and Cauchy trajectory in Riemannian manifolds. First we prove that under rather general hypotheses the central path defined by a general barrier for a monotone variational inequality problem is well defined, bounded, and continuous and converges to the analytic center of the solution set (with respect to the given barrier), thus generalizing results which deal only with complementarity problems and with the logarithmic barrier. Next we prove that a sequence generated by the proximal point method with the Bregman distance naturally induced by the barrier function converges precisely to the same point. Furthermore, for a certain class of problems (including linear programming), such a sequence is contained in the central path, making the concepts of central path and generalized proximal point sequence virtually equivalent. Finally we prove that for this class of problems the central path also coincides with the Cauchy trajectory in the Riemannian manifold defined on the positive orthant by a metric given by the Hessian of the barrier (i.e., a curve whose direction at each point is the negative gradient of the objective function at that point in the Riemannian metric).
This paper introduces the concept of exceptional family for nonlinear variational inequality problems. Among other things, we show that the nonexistence of an exceptional family is a sufficient condition for the exist...
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This paper introduces the concept of exceptional family for nonlinear variational inequality problems. Among other things, we show that the nonexistence of an exceptional family is a sufficient condition for the existence of a solution to variational inequalities. This sufficient condition is weaker than many known solution conditions and it is also necessary for pseudomonotone variational inequalities. From the results in this paper, we believe that the concept of exceptional families of variational inequalities provides a new powerful tool for the study of the existence theory for variational inequalities.
We study the well definedness of the central path for a linearly constrained convex programming problem with smooth objective function. We prove that, under standard assumptions, existence of the central path is equiv...
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We study the well definedness of the central path for a linearly constrained convex programming problem with smooth objective function. We prove that, under standard assumptions, existence of the central path is equivalent to the nonemptiness and boundedness of the optimal set. Other equivalent conditions are given. We show that, under an additional assumption on the objective function, the central path converges to the analytic center of the optimal set.
We treat in this paper linear programming (LP) problems with uncertain data. The focus is on uncertainty associated with hard constraints: those which must be satisfied, whatever is the actual realization of the data ...
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We treat in this paper linear programming (LP) problems with uncertain data. The focus is on uncertainty associated with hard constraints: those which must be satisfied, whatever is the actual realization of the data (within a prescribed uncertainty set). We suggest a modeling methodology whereas an uncertain LP is replaced by its robust counterpart (RC). We then develop the analytical and computational optimization tools to obtain robust solutions of an uncertain LP problem via solving the corresponding explicitly stated convex RC program. In particular, it is shown that the RC of an LP with ellipsoidal uncertainty set is computationally tractable, since it leads to a conic quadratic program, which can be solved in polynomial time. (C) 1999 Published by Elsevier Science B.V. All rights reserved.
An optimization problem with an affine feasible set is studied. By converting the affine set to a closed convex set which contains the solution, the Meyer-Polak algorithm can be used. The selection of the constraints ...
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An optimization problem with an affine feasible set is studied. By converting the affine set to a closed convex set which contains the solution, the Meyer-Polak algorithm can be used. The selection of the constraints is the fey. issue. A theorem has been derived for the selection of the constraints for the affine problem where the cost function is of p-norm form, Its application in a control problem is demonstrated.
In this paper we present several "infeasible-start" path-following and potential-reduction primal-dual interior-point methods for nonlinear conic problems. These methods try to find a recession direction of ...
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In this paper we present several "infeasible-start" path-following and potential-reduction primal-dual interior-point methods for nonlinear conic problems. These methods try to find a recession direction of the feasible set of a self-dual homogeneous primal-dual problem. The methods under consideration generate an epsilon-solution for an epsilon-perturbation of an initial strictly (primal and dual) feasible problem in O(root nu ln nu/epsilon rho(f)) iterations, where nu is the parameter of a self-concordant barrier for the cone, epsilon is a relative accuracy and rho(f) is a feasibility measure. We also discuss the behavior of path-following methods as applied to infeasible problems. We prove that strict infeasibility (primal or dual) can be detected in O(root nu ln nu/rho) iterations, where rho. is a primal or dual infeasibility measure.
We give a method for minimizing a convex function f that generates a sequence {x(k)} by taking x(k) to be an approximate minimizer of f(k) + D-h(., xk(-1))/t(k), where f(k) is a piecewise linear model of f constructed...
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We give a method for minimizing a convex function f that generates a sequence {x(k)} by taking x(k) to be an approximate minimizer of f(k) + D-h(., xk(-1))/t(k), where f(k) is a piecewise linear model of f constructed from accumulated subgradient linearizations of f, Dh is the D-function of a generalized Bregman function h and t(k) > 0. Convergence under implementable criteria is established by extending our recent framework of Bregman proximal minimization, which is of independent interest, e.g., for nonquadratic multiplier methods for constrained minimization. In particular, we provide new insights into the convergence properties of bundle methods based on h = 1/2\.\(2).
Asymptotic necessary and sufficient conditions for a point to be a Pareto minimum, and weak minimum (proper minimum) for a convex multi-objective program are given without a regularity condition. It is further shown t...
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Asymptotic necessary and sufficient conditions for a point to be a Pareto minimum, and weak minimum (proper minimum) for a convex multi-objective program are given without a regularity condition. It is further shown that, in the cases of weak minimum and single objective function, the asymptotic dual conditions reduce to nonasymptotic optimality conditions under Slater's constraint qualification. The results are applied to multi-objective quadratic and linar programming problems. Numerical examples are given to illustrate the nature of the conditions.
A new approximate proximal point method for minimizing the sum of two convex functions is introduced. It replaces the original problem by a sequence of regularized subproblems in which the functions are alternately re...
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A new approximate proximal point method for minimizing the sum of two convex functions is introduced. It replaces the original problem by a sequence of regularized subproblems in which the functions are alternately represented by linear models. The method updates the linear models and the prox center, as well as the prox coefficient. It is monotone in terms of the objective values and converges to a solution of the problem, if any. A dual version of the method is derived and analyzed. Applications of the methods to multistage stochastic programming problems are discussed and preliminary numerical experience is presented.
In this note we give some sharp estimates for norms of polynomials via the products of norms of their linear terms. Different convex norms on the unit disc are considered.
In this note we give some sharp estimates for norms of polynomials via the products of norms of their linear terms. Different convex norms on the unit disc are considered.
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