Showing papers on "Convex optimization published in 1972"
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01 Dec 1972
TL;DR: In this article, the authors propose a method to solve the problem of convex control problems in Banach spaces. But this method is not suitable for functional analysis.Convex Functions and Convex Programming
Abstract: Fundamentals of Functional Analysis.- Convex Functions.- Convex Programming.- Convex Control Problems in Banach Spaces.
853 citations
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TL;DR: A family of real functions, called r-convex functions, which represents a generalization of the notion of convexity is introduced, and is included in the family of quasiconveX functions.
Abstract: A family of real functions, calledr-convex functions, which represents a generalization of the notion of convexity is introduced This family properly includes the family of convex functions and is included in the family of quasiconvex functions Some properties ofr-convex functions are derived and relations with other generalizations of convex functions are discussed
126 citations
99 citations
TL;DR: In this article, a method of computing solutions of variational inequalities in a finite-dimensional space was proposed. But this method is quite close to the method of Theil-Van de Panne described in Ref. 1 in the case of quadratic programming.
Abstract: We observe that variational inequalities generalize convex programming. We look here for a method of computing solutions of variational inequalities in a finite-dimensional space. The method we propose is quite close to the method of Theil-Van de Panne described in Ref. 1 in the case of quadratic programming.
88 citations
01 Dec 1972
TL;DR: A new penalty function algorithm of this type is presented which offers significant advantages over existing schemes for the case of the convex programming problem and treats inequality constraints explicitly.
Abstract: During recent years it has been shown that the performance of penalty function methods for constrained minimization can be improved significantly by introducing gradient type iterations for solving the dual problem. In this paper we present a new penalty function algorithm of this type which offers significant advantages over existing schemes for the case of the convex programming problem. The algorithm treats inequality constraints explicitly and can also be used for the solution of general mathematical programming problems.
83 citations
TL;DR: Two upper bounds on the error incurred by using the approximation are developed and the formulation can be used in convex programming to solve some nonlinearly constrained problems.
Abstract: This paper provides a method for approximating optimal location in a multi-facility Weber problem where rectangular distances apply. Optimality is achieved when the sum of weighted distances is minimized. Two upper bounds on the error incurred by using the approximation are developed. The formulation can be used in convex programming to solve some nonlinearly constrained problems.
74 citations
TL;DR: In this article, the pertinence of convexity arguments in the study of discrepancy of sequences is exhibited, and the usefulness of this viewpoint can be twofold: it allows the interpretation of the problem of estimating the discrepancy as a problem in convex programming in important cases, and it helps to restrict the family of sets which have to be considered when evaluating the usual (or extreme) discrepancy and the isotrope discrepancy of sequence.
Abstract: The pertinence of convexity arguments in the study of discrepancy of sequences is exhibited. The usefulness of this viewpoint can be twofold. Firstly, it allows the interpretation of the problem of estimating the discrepancy as a problem in convex programming in important cases. Secondly, it helps to restrict the family of sets which have to be considered when evaluating the usual (or extreme) discrepancy and the isotrope discrepancy of sequences. In particular, in the latter case it suffices to look at a rather special class of convex polytopes.
57 citations
TL;DR: In this paper, the problem of selecting an optimal mix of pipe diameters and slopes, given a set of economic and technological inputs as well as a network layout, is formulated as a separable convex programming problem.
Abstract: This paper addresses itself to the question of obtaining the minimum cost design for a wastewater collection network. The concept of optimization is explored with respect to overall collection networks. Present design methodologies and recent developments in both network layout and design are explored. The design problems of selecting an optimal mix of pipe diameters and slopes, given a set of economic and technological inputs as well as a network layout is then formulated as a separable convex programming problem. The formulation guarantees the generation of a global optimal solution, and a numerical solution can be obtained using existing commercially available computer software. The paper concludes with an evaluation of the developed procedure and its possible adoption in everyday sewer design.
36 citations
TL;DR: In this paper, a generalization of Farkas' lemma to nonlinear functions and to infinite-dimensional spaces is presented, and the necessary condition for constrained minima is deduced for infinite dimension and cone constraints.
Abstract: Farkas' lemma is generalized both to nonlinear functions and to infinite-dimensional spaces; the version for linear maps is less restricted than Hurwicz's result. A generalization of F. John's necessary condition for constrained minima is deduced for infinite dimension and cone constraints. Some theorems on converse and symmetric duality in nonlinear programming are obtained, which extend the known results, even in the finite-dimensional case.
TL;DR: This work generalizes Martos' and Koecher's results in [12] and [13] by using Koechers' results in ['8], which describe quasi-convex and pseudo-conventus quadratic functions on solid convex sets.
Abstract: We study quasi-convex and pseudo-convex quadratic functions on solid convex sets. This generalizes Martos' results in [12] and [13] by using Koecher's results in [8].
TL;DR: For the minimum problem of the functional (where, and the case corresponds to some constraints imposed on and ) as discussed by the authors consider the existence of a function which has the following property: if t) is a minimizing sequence, then, for any and which, and for any, (every function that has this property yields a necessary condition for the absolute minimum).
Abstract: For the minimum problem of the functional (where , and the case corresponds to some constraints imposed on and ) we consider the problem of the existence of a function which has the following property: if t) is a minimizing sequence, then, for any and which , and for any , (every function which has this property yields a necessary condition for the absolute minimum). We prove existence criterions for an arbitrary and continuous function .Bibliography: 9 items.
TL;DR: In this paper, a representation theorem for the tensor product of locally convex modules is presented, which has a number of consequences in the study of the multiplier problem in harmonic analysis, and the remainder of the paper is devoted to these applications.
Abstract: In this paper we present a representation theorem for the tensor product of locally convex modules. This theorem has a number of consequences in the study of the multiplier problem in harmonic analysis, and the remainder of the paper is devoted to these applications.
TL;DR: In this paper, the authors formulate a terminal control problem and present its solution, which is obtained from the results of a minimization of a convex function that is rapidly solved by a computer.
Abstract: In this paper, we formulate a terminal control problem and present its solution. The solution is obtained from the results of a minimization of a convex function that is rapidly solved by a computer. It is shown that a number of other problems considered in the literature can be given in this formulation and, consequently, computational solutions for these problems can be obtained in a straightforward manner
TL;DR: In this article, the following result is proved: there is an entire function of finite order such that its indicator and its lower indicator can be found in a convex function, and applications of this theorem are given.
Abstract: In this paper, the following result is proved.Theorem. Let and be two -trigonometrically convex functions. There is an entire function of finite order such that its indicator and its lower indicator . Applications of this theorem are given.Bibliography: 6 items.
01 Jan 1972