We consider the volumetric cutting plane method for finding a point in a convex set C subset of R-n that is characterized by a separation oracle. We prove polynomiality of the algorithm with each added cut placed dire...
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We consider the volumetric cutting plane method for finding a point in a convex set C subset of R-n that is characterized by a separation oracle. We prove polynomiality of the algorithm with each added cut placed directly through the current point and show that this "central cut" version of the method can be implemented using no more than 25n constraints at any time.
In this paper, an unconstrained convex programming dual approach for solving a class of linear semi-infinite programming problems is proposed. Both primal and dual convergence results are established under some basic ...
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In this paper, an unconstrained convex programming dual approach for solving a class of linear semi-infinite programming problems is proposed. Both primal and dual convergence results are established under some basic assumptions. Numerical examples are also included to illustrate this approach.
In this report the problem of minimization of a convex function f ( x ) on a convex closed and bounded set Q ⊂ R n is considered. The method described below concerns gradient methods of the search of extremum of conve...
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In this report the problem of minimization of a convex function f ( x ) on a convex closed and bounded set Q ⊂ R n is considered. The method described below concerns gradient methods of the search of extremum of convex functions. The discrete analogue of the approximation gradient plays here the role of the gradient (Batukhtin and Maiboroda, 1984; Batukhtin and Maiboroda, 1995).
The aim of this paper is to carry out an exhaustive post optimization analysis in a convex Goal programming problem, so as to study the possible existence of satisfying solutions for different levels of the target val...
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The aim of this paper is to carry out an exhaustive post optimization analysis in a convex Goal programming problem, so as to study the possible existence of satisfying solutions for different levels of the target values. To this end, an interactive algorithm is proposed, which allows us to improve the values of the objective functions, after obtaining a satisfying solution, if such a solution exists, in such a way that a Pareto optimal solution is finally reached, through a successive actualization of such target values. This way, the target values are lexicographically improved, according to the priority order previously given by the decision maker, in an attempt to harmonize the concepts of satisfying and efficient solutions, which have traditionally been in conflict. (C) 1998 Elsevier Science B.V.
This paper presents a new interior point algorithrn for linearly constrainedconvex programming which is based upon interior ellipsoid mehtod- It is shown thatthe method is a polynomial time algorithm.
This paper presents a new interior point algorithrn for linearly constrainedconvex programming which is based upon interior ellipsoid mehtod- It is shown thatthe method is a polynomial time algorithm.
We give efficiency estimates for proximal bundle methods for finding f* := min(x)f, where f and X are convex. We show that, for any accuracy epsilon>0, these methods find a point x(k) is an element of X such that f...
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We give efficiency estimates for proximal bundle methods for finding f* := min(x)f, where f and X are convex. We show that, for any accuracy epsilon>0, these methods find a point x(k) is an element of X such that f(x(k)) - f* less than or equal to epsilon after at most k = O (1/epsilon(3)) objective and subgradient evaluations.
A general approach to analyze convergence of the proximal-like methods for variational inequalities with set-valued maximal monotone operators is developed. It is oriented to methods coupling successive approximation ...
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A general approach to analyze convergence of the proximal-like methods for variational inequalities with set-valued maximal monotone operators is developed. It is oriented to methods coupling successive approximation of the variational inequality with the proximal point algorithm as well as to related methods using regularization on a subspace and weak regularization. This approach also covers so-called multistep regularization methods, in which the number of proximal iterations in the approximated problems is controlled by a criterion characterizing these iterations as to be effective. The conditions on convergence require control of the exactness of the approximation only in a certain region of the original space. Conditions ensuring linear convergence of the methods are established.
Routing problems appear frequently when dealing with the operation of communication or transportation networks. Among them, the message routing problem plays a determinant role in the optimization of network performan...
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Routing problems appear frequently when dealing with the operation of communication or transportation networks. Among them, the message routing problem plays a determinant role in the optimization of network performance. Much of the motivation for this work comes from this problem which is shown to belong to the class of nonlinear convex multicommodity flow problems. This paper emphasizes the message routing problem in data networks, but it includes a broader literature overview of convex multicommodity flow problems. We present and discuss the main solution techniques proposed for solving this class of large-scale convex optimization problems. We conduct some numerical experiments on the message routing problem with four different techniques.
We apply a recent extension of the Bregman proximal method for convex programming to LP relaxations of 0-1 problems. We allow inexact subproblem solutions obtained via dual ascent, increasing their accuracy successive...
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We apply a recent extension of the Bregman proximal method for convex programming to LP relaxations of 0-1 problems. We allow inexact subproblem solutions obtained via dual ascent, increasing their accuracy successively to retain global convergence. Our framework is applied to relaxations of large-scale set covering problems that arise in airline crew scheduling. Approximate relaxed solutions are used to construct primal feasible solutions via a randomized heuristic. Encouraging preliminary experience is reported.
We consider the forward-backward splitting method for finding a zero of the sum of two maximal monotone mappings. This method is known to converge when the inverse of the forward mapping is strongly monotone. We propo...
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We consider the forward-backward splitting method for finding a zero of the sum of two maximal monotone mappings. This method is known to converge when the inverse of the forward mapping is strongly monotone. We propose a modification to this method, in the spirit of the extragradient method for monotone variational inequalities, under which the method converges assuming only the forward mapping is (Lipschitz) continuous on some closed convex subset of its domain. The modification entails an additional forward step and a projection step at each iteration. Applications of the modified method to decomposition in convex programming and monotone variational inequalities are discussed.
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