Showing papers on "Convex optimization published in 1982"
TL;DR: Three measures of divergence between vectors in a convex set of a n -dimensional real vector space are defined in terms of certain types of entropy functions, and their convexity property is studied.
Abstract: Three measures of divergence between vectors in a convex set of a n -dimensional real vector space are defined in terms of certain types of entropy functions, and their convexity property is studied. Among other results, a classification of the entropies of degree \alpha is obtained by the convexity of these measures. These results have applications in information theory and biological studies.
437 citations
TL;DR: The Kuhn-Tucker conditions for the existence of solutions to inequality constrained minimization problems are sufficient conditions when all the functions involved are convex as discussed by the authors, and various concepts such as pseudo-convexity and quasi-concaveity have been introduced for the purpose of weakening this limitation of convexity in mathematical programming.
Abstract: The Kuhn-Tucker conditions for the existence of solutions to inequality constrained minimization problems are sufficient conditions when all the functions involved are convex. Various concepts such as pseudo-convexity and quasi-convexity have been introduced for the purpose of weakening this limitation of convexity in mathematical programming. Still wider classes of functions are introduced in this paper and applied in Kunh-Tucker theory and in duality theory.
210 citations
Book•
01 Jan 1982
133 citations
01 Jan 1982
98 citations
TL;DR: Comparing the performance of several existing methods for determining the equilibrium network flows of a small realistic network model in which intersection controls are incorporated which lead to particularly asymmetric cost functions is compared.
Abstract: We consider algorithms proposed for solving the fixed demand user optimized network equilibrium problem with asymmetric user costs. Because the Jacobian matrix for the costs is asymmetric, no known equivalent convex optimization problem exists and alternative solution methods to nonlinear programming techniques must be sought. The purpose of this paper is to compare the performance of several existing methods for determining the equilibrium network flows of a small realistic network model in which intersection controls are incorporated which lead to particularly asymmetric cost functions.
88 citations
TL;DR: In this article, the problem of multidimensional maximum entropy method (MEM) spectral estimation from nonuniformly spaced correlation measurements is investigated and a necessary and sufficient condition is derived for the existence and uniqueness of the MEM spectral estimate in its usual form.
Abstract: The problem of multidimensional maximum entropy method (MEM) spectral estimation from nonuniformly spaced correlation measurements is investigated. A necessary and sufficient condition is derived for the existence and uniqueness of the MEM spectral estimate in its usual form. It is shown that this condition is not satisfied in many multidimensional problems of interest, although it is satisfied in the important practical case of spectral supports composed of a finite number of points. When the existence condition is satisfied, calculation of the MEM estimate reduces to the solution of a finite-dimensional convex optimization problem. The application of standard optimization techniques to this problem results in iterative computational algorithms which are guaranteed to converge. The algorithms so obtained are compared to those previously proposed and a spectral estimation example is presented.
82 citations
TL;DR: Sklansky's definition of digital convexity is equivalent to other definitions under new schemes for digitizing regions and arcs and a linear time algorithm is presented that determines the smallest integer n such that the region is adigital convex n-gon.
Abstract: New schemes for digitizing regions and arcs are introduced. It is then shown that under these schemes, Sklansky's definition of digital convexity is equivalent to other definitions. Digital convex polygons of n vertices are defined and characterized in terms of geometric properties of digital line segments. Also, a linear time algorithm is presented that, given a digital convex region, determines the smallest integer n such that the region is a digital convex n-gon.
63 citations
48 citations
TL;DR: In this article, necessary and sufficient conditions for a nonconvex quadratic function to take a local minimum over a convex set are provided, and various limiting examples are given.
Abstract: We provide necessary and sufficient conditions for a (non-convex) quadratic function to take a local minimum over a convex set. Various limiting examples are given.
43 citations
TL;DR: It is proved that a digital solid is convex if and only if it has the chordal triangle property, which is only necessary, but not sufficient, conditions for a digitalSolid to be convex.
Abstract: A definition of convexity of digital solids is introduced. Then it is proved that a digital solid is convex if and only if it has the chordal triangle property. Other geometric properties which characterize convex digital regions are shown to be only necessary, but not sufficient, conditions for a digital solid to be convex. An efficient algorithm that determines whether or not a digital solid is convex is presented.
43 citations
TL;DR: The concern is with solving as linear or convex quadratic programs special cases of the optimal containment and meet problems of a set for which some translation contains a set or meets each element in a collection of sets, respectively.
Abstract: The concern is with solving as linear or convex quadratic programs special cases of the optimal containment and meet problems. The optimal containment or meet problem is that of finding the smallest scale of a set for which some translation contains a set or meets each element in a collection of sets, respectively. These sets are unions or intersections of cells where a cell is either a closed polyhedral convex set or a closed solid ball.
TL;DR: Duality results are established in convex programming with the set-inclusive constraints studied by Soyster and duality theory for generalized linear programs by Thuente is further generalized.
Abstract: Duality results are established in convex programming with the set-inclusive constraints studied by Soyster. The recently developed duality theory for generalized linear programs by Thuente is further generalized and also brought into the framework of Soyster's theory. Convex programming with set-inclusive constraints is further extended to fractional programming.
TL;DR: Based upon a theorem for lower and upper bounds on the Lagrange multiplier a fully polynomial time approximation scheme is proposed and the efficiency of the algorithm is demonstrated by a computational experiment.
TL;DR: It is shown that convexity is necessary, and a similar theorem is proved for stationary points when the functions are not necessarily convex but the gradient exists for each function.
Abstract: We give a short proof that in a convex minimax optimization problem ink dimensions there exist a subset ofk + 1 functions such that a solution to the minimax problem with thosek + 1 functions is a solution to the minimax problem with all functions. We show that convexity is necessary, and prove a similar theorem for stationary points when the functions are not necessarily convex but the gradient exists for each function.
TL;DR: In this paper, the authors introduced a new class of generalized convex functions and showed that a real function f which is continuous on a compact convex subset M of R /sub n/ and whose set of global minimizers on M is arcwise-connected has the property that every local minimum is global if, and only if, f belongs to that class of functions.
Abstract: In this note, we introduce a new class of generalized convex functions and show that a real function f which is continuous on a compact convex subset M of R /sub n/ and whose set of global minimizers on M is arcwise-connected has the property that every local minimum is global if, and only if, f belongs to that class of functions.
01 Mar 1982
TL;DR: A priori information in the form that the solution lies in a convex subset of a closed linear manifold, is assumed in this article, where theoretical and experimental results are included.
Abstract: : This report treats in the problem of iterative image restoration from projection operators in an infinite dimensional Hilbert Space. A priori information in the form that the solution lies in a convex subset of a closed linear manifold, is assumed. Theoretical and experimental results are included. (Author)
TL;DR: In this paper, the authors consider the problem of finding the maximum likelihood estimate for a certain class of discrete sampling models using convex optimization theory, including aspects of equivalence theory, duality theory, and iterative procedures.
Abstract: In this article we consider the pioblem of finding the maximum likelihood estimate for a certain class of discrete sampling models. Methods are adapted from parts of convex optimization theory, includiug aspects of equivalence theory, duality theorv and iterative procedures Their application is illustrated through example.
TL;DR: In this article, the authors consider a model of capital accumulation and prove the existence of a support price path for the optimal path of the capital accumulation in a continuous time model of infinite horizon.
Abstract: We consider a model of capital accumulation and prove the existence of a support price path for the optimal path of capital accumulation. The considered model is a continuous time model of infinite horizon. Our problem is the so-called convex problem of optimal control without differentiability. We adopt the overtaking optimality criterion and prove the existence of a dual price path which supports the value function as well as the Hamiltonian function.
01 Jan 1982
TL;DR: In this paper, the error bounds for all components of the inverse of the Hilbert 15 × 15 matrix are as small as possible, that is, left and right bounds differ only by one in the 12 place of the mantissa of each component.
Abstract: Publisher Summary This chapter presents the new methods of solving algebraic problems with high accuracy. Examples of such problems are the solving of linear systems, eigenvalue/eigenvector determination, computing zeros of polynomials, sparse matrix problems, computation of the value of an arbitrary arithmetic expression, in particular, the value of a polynomial at a point, nonlinear systems, linear, quadratic, and convex programming over the field of real or complex numbers as well as over the corresponding interval spaces. All the algorithms based on new methods have some key properties in common: (1) every result is automatically verified to be correct by the algorithm; (2) the results are of high accuracy, that is, the error of every component of the result is of the magnitude of the relative rounding error unit; (3) the solution of the given problem is automatically shown to exist and to be unique within the given error bounds; and (4) the computing time is of the same order as comparable floating-point algorithm. The key property of the algorithms is that error control is performed automatically by the computer without any requirement on the part of the user, such as estimating spectral radii. The error bounds for all components of the inverse of the Hilbert 15 × 15 matrix are as small as possible, that is, left and right bounds differ only by one in the 12 place of the mantissa of each component. It is called least significant bit accuracy.
TL;DR: An apparently novel way of constructing the subgradient of a convex function defined on a finite dimensional vector space is described.
Abstract: We describe an apparently novel way of constructing the subgradient of a convex function defined on a finite dimensional vector space.
TL;DR: Gauthier-Villars as discussed by the authors implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions).
Abstract: © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1982, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
TL;DR: In this paper, a K-function is constructed for quasi-convex optimization problems and connections between the following generalizations of convex conjugate functions and corresponding sets are given.
Abstract: In this paper some connections between the following generalizations of convex conjugate functions and corresponding sets are given: quasi-conjugate functions, φ-conjugate functionsK-functions, convexity properties of set-families, Especially, a K-function is constructed for quasi-convex optimization problems.
TL;DR: In this article, it was shown that differentiability is a generic property of convex functions and not a special case of differentiability in convex convex function functions, i.e.
TL;DR: In this article, necessary and sufficient optimality conditions of Kuhn-Tucker type for a convex programming problem with subdifferentiable operator constraints have been obtained, and a duality theorem of Wolfe's type has been derived.
Abstract: Necessary and sufficient optimality conditions of Kuhn-Tucker type for a convex programming problem with subdifferentiable operator constraints have been obtained. A duality theorem of Wolfe's type has been derived. Assuming that the objective function is strictly convex, a converse duality theorem is obtained. The results are then applied to a programming problem in which the objective function is the sum of a positively homogeneous, lower-semi-continuous, convex function and a continuous convex function.
TL;DR: In this paper, it was shown that for 1≤p < ∞ there exist generalized convex functions for which uniqueness of best Lp-approxima-tions from Sn,k fails.
Abstract: Several authors made contributions to the result that for 1≤p≤∞ the function has a unique best Lp-approximation from Sn, k, the set of spline functions of degree n with k free knots. Jetter and Lange [Die Eindeutigkeit L2-optimaier polynomialer Monosplines, Math. Z. 158 (178), 23-34] conjectured that for p=2 this result remains valid for all generalized convex functions ff i.e [a,b] and f (n+1)>o on [a,b]. However, our main result says that for 1≤p<∞ there exist generalized convex functions for which uniqueness of best Lp-approxima- tions from Sn,k fails.
TL;DR: In this paper, the authors define a family of semi-infinite minimization problems with family parameter and parameter space and prove that if the minimum set mapping is upper semicontinuous, then at least one minimum point satisfies the criterion.
Abstract: Publisher Summary This chapter discusses the parametric approximation and optimization and defines a family of semi-infinite minimization problems with family parameter and parameter space. A parametric linear optimization problem is described with variable matrix and variable restriction vector. Parametric linear finite optimization has many applications. The chapter proves an always sufficient criterion for a minimal point and introduces pointwise convex optimization problem. Many important optimization problems are pointwise convex, for example linear, convex, and fractional optimization problems. It is shown that if the minimum set mapping is upper semicontinuous, then at least one minimum point satisfies the criterion.