Showing papers on "Convex optimization published in 1985"

Methods of Descent for Nondifferentiable Optimization

Abstract: Fundamentals.- Aggregate subgradient methods for unconstrained convex minimization.- Methods with subgradient locality measures for minimizing nonconvex functions.- Methods with subgradient deletion rules for unconstrained nonconvex minimization.- Feasible point methods for convex constrained minimization problems.- Methods of feasible directions for nonconvex constrained problems.- Bundle methods.- Numerical examples.

Generalized Differentiability / Duality and Optimization for Problems Dealing with Differences of Convex Functions

Abstract: A function is called d. c. if it can be. expressed as a difference of two convex functions. In the present paper we survey the main known results about suuch functions from the viewpoint of Analysis and Optimization.

Finite Algorithms in Optimization and Data Analysis

Abstract: Preface Table of Notation Some Results from Convex analysis Linear Programming Applications of linear Programming in Discrete Approximation Polyhedral Convex Functions Least Squares and Related Methods Some Applications to Non-Convex Problems Some Questions of Complexity and Performance Appendices References Index.

Applications of the method of partial inverses to convex programming: Decomposition

Abstract: A primal–dual decomposition method is presented to solve the separable convex programming problem Convergence to a solution and Lagrange multiplier vector occurs from an arbitrary starting point The method is equivalent to the proximal point algorithm applied to a certain maximal monotone multifunction In the nonseparable case, it specializes to a known method, the proximal method of multipliers Conditions are provided which guarantee linear convergence

Signal synthesis in the presence of an inconsistent set of constraints

Abstract: In this paper, we present a novel technique for signal synthesis in the presence of an inconsistent set of constraints. This technique represents a general, minimum norm, solution to the class of synthesis problems in which: the desired signal may be characterized as being an element of some Hilbert Space; each of the N design constraints generates a closed convex set in that space; and those N convex sets generate, or may be resolved into, two disjoint closed convex sets, such that at least one of the two sets is bounded. The synthesis technique employs alternating nearest point maps onto closed convex subsets of a Hilbert Space, and may be viewed as an extension of D. Youla's "Method of Convex Projections"--which addresses the case in which the N closed convex sets, corresponding to the design constraints, possess a nonempty intersection. Section I provides a general introduction to the synthesis problem and to its solution. Section II contains the mathematical justification for the solution technique, while Section III presents an example of the synthesis of a data window for spectral estimation. In Section IV, we discuss potential extensions of this technique within the area of signal synthesis, as well as to the more general class of constrained optimization problems.

A Conical Algorithm for Globally Minimizing a Concave Function Over a Closed Convex Set

Abstract: In this paper we are concerned with the problem of finding the global minimum of a concave function over a closed, convex, possibly unbounded set in Rn. The intrinsic difficulty of this problem is due to the fact that a local minimum of the objective function may fail to be a global one—which makes the conventional methods of local optimization almost useless.

Lower subdifferentiable functions and their minimization by cutting planes

Abstract: This paper introduces lower subgradients as a generalization of subgradients. The properties and characterization of boundedly lower subdifferentiable functions are explored. A cutting plane algorithm is introduced for the minimization of a boundedly lower subdifferentiable function subject to linear constraints. Its convergence is proven and the relation is discussed with the well-known Kelley method for convex programming problems. As an example of application, the minimization of the maximum of a finite number of concave-convex composite functions is outlined.

A Condition Number for Differentiable Convex Inequalities

Abstract: For a system of differentiable convex inequalities, a new bound is given for the absolute error in an infeasible point in terms of the absolute residual. By using this bound a condition number is defined for the system of inequalities which gives a bound for the relative error in an infeasible point in terms of the relative residual.

Smoothingq-convex functions and vanishing theorems

Abstract: In [1] Andreotti and Grauert showed that the existence of a C ~ (strongly) qconvex resp. (strongly) q-concave exhaustion function on a complex space implies certain cohomology finiteness and vanishing theorems. However, in some important concrete situations, where one would like to apply the results of Andreotti-Grauert , the following difficulty arises: The natural exhaustion functions which appear are not C ~, but are near each given point only the supremum of a finite number of strongly q-convex (resp. the infimum of a finite number of strongly q-concave) C ~ functions. The most simple examples of this kind are, of course, finite intersections of strongly q-convex (resp. finite unions of strongly q-concave) domains. Perhaps the most important class of examples arises as complements of algebraic subvarieties of IP\" (see theorem below). It is, therefore, of considerable interest to study the influence which the existence of such merely continuous q-convex (resp. q-concave) exhaustion functions has on the cohomology of the spaces. Such a study was done for the first time by Grauert in his article \"Kantenkohomologie\" (the superlevel sets of functions which are locally the infimum of a finite number of C ~ functions have more or less complicated corners, \"Kanten\"). In Satz 1 of this article it is assumed, that a domain B is given on a neighborhood W of a point Zoe~?B as

On the Solution of Variational Inequalities by the Ellipsoid Method

Abstract: Variational inequalities provide a convenient mathematical approach for unifying results relating to extremum and equilibrium problems. In this paper we demonstrate the applicability of the ellipsoid method to solve monotone variational inequalities. A convergence proof is given, which, for example, in case of convex optimization explains the linear convergence of the record points. We compare this result with Goffin's convergence proof by extending his proof technique to the constrained case in the appendix.

Differentiability with respect to parameters of solutions to convex programming problems

Abstract: We consider a family of convex programming problems that depend on a vector parameter, characterizing those values of parameters at which solutions and associated Lagrange multipliers are Gâteaux differentiable. These results are specialized to the problem of the metric projection onto a convex set. At those points where the projection mapping is not differentiable the form of Clarke's generalized derivative of this mapping is derived.

An algorithm for nonsmooth convex minimization with errors

Abstract: A readily implementable algorithm is given for minimizing any convex, not necessarily differentiable, function f of several variables. At each iteration the method requires only one approximate evaluation of f and its e-subgradient, and finds a search direction by solving a small quadratic programming problem. The algorithm generates a minimizing sequence of points, which converges to a solution wheneverf has any minimizers.

Simple bounds for solutions of monotone complementarity problems and convex programs

Abstract: For a solvable monotone complementarity problem we show that each feasible point which is not a solution of the problem provides simple numerical bounds for some or all components of all solution vectors. Consequently for a solvable differentiable convex program each primal-dual feasible point which is not optimal provides simple bounds for some or all components of all primal-dual solution vectors. We also give an existence result and simple bounds for solutions of monotone compementarity problems satisfying a new, distributed constraint qualification. This result carries over to a simple existence and boundedness result for differentiable convex programs satisfying a similar constraint qualification.

Continuous algorithms for solution of convex optimization problems and finding saddle points of contex-coneave functions with the use of projection operations

Abstract: Continuous algorithms are proposed and studied for solution of convex programming problems and for finding the saddle points of convex-concave functions by the use of projection. The theory of mono...

A convex-like duality scheme for quasi-convex programs

Abstract: This paper describes a symmetric duality relation for quasi-convex programs. We are able to strengthen previous results and to define necessary and sufficient conditions for the absence of duality gap. In the present scheme one can generate quasi-convex quasi-concave Lagrangians and discuss the correspondence between saddle points of the Lagrangians and the solutions to the dual and primal programs. The present scheme is very similar to Rockafellar's scheme for convex programs and in this sense it may be viewed as a unified approach. Several examples are also given.

Approximate solutions of vector optimization problems

Abstract: In this paper we introduce different notions of approximate extremal points of sets in ordered vector spaces. Following this we state duality relationships related to these types of approximate solutions of vector valued convex optimization problems, and we also mention some applications of these results. Finally we present a vector valued version of Ekeland's famous variational principle, a tool with so many applications in scalar optimization.

Controllability of convex processes

Abstract: The purpose of this paper is to provide several characterizations of controllability of differential inclusions whose right-hand sides are convex processes. Convex processes are the set-valued maps whose graphs are convex cones; they are the set-valued analogues of linear operators. Such differential inclusions include linear systems where the controls range over a convex cone (and not only a vector space). The characteristic properties are couched in terms of invariant cones by convex processes, or eigenvalues of convex processes, or a rank condition. We also show that the controllability is equivalent to the observability of the adjoint inclusion.

Nonlinear Programming Techniques Applied to Stochastic Programs with Recourse

Abstract: Stochastic convex programs with recourse can equivalently be formulated as nonlinear convex programming problems. These possess some rather marked characteristics. Firstly, the proportion of linear to nonlinear variables is often large and leads to a natural partition of the constraints and objective. Secondly, the objective function corresponding to the nonlinear variables can vary over a wide range of possibilities; under appropriate assumptions about the underlying stochastic program it could be, for example, a smooth function, a separable polyhedral function or a nonsmooth function whose values and gradients are very expensive to compute. Thirdly, the problems are often large-scale and linearly constrained with special structure in the constraints. This paper is a comprehensive study of solution methods for stochastic programs with recourse viewed from the above standpoint. We describe a number of promising algorithmic approaches that are derived from methods of nonlinear programming. The discussion is a fairly general one, but the solution of two classes of stochastic programs with recourse are of particular interest. The first corresponds to stochastic linear programs with simple recourse and stochastic right-hand-side elements with given discrete probability distribution. The second corresponds to stochastic linear programs with complete recourse and stochastic right-hand-side vectors defined by a limited number of scenarios, each with given probability. A repeated theme is the use of the MINOS code of Murtagh and Saunders as a basis for developing suitable implementations.

The General Concept of Cone Approximations in Nondifferentiable Optimization

Abstract: General optimization problems connected with necessary conditions for optimality have been studied by many authors in recent years. Since Clarke (1975) introduced the notion of a generalized gradient and the corresponding tangent cone, numerous papers have been published which extend standard smooth and convex optimization results to the general case.

On the continuity of the minima for a family of constrained optimization problems

Abstract: We consider a family of problems Py dealing with the minimization of a given function on a constraint set, both depending on a parameter y. We study continuity properties, with respect to a parameter, of the value and of the solution set of the problems. Working with convex functions and convex constraint sets, we show how the well-posedness of the problem allows to avoid compactness hypotheses usually requested to get the same stability results.

Quadratic geometric programming with application to machining economics

Abstract: Geometric Programming is extended to include convex quadratic functions. Generalized Geometric Programming is applied to this class of programs to obtain a convex dual program. Machining economics problems fall into this class. Such problems are studied by applying this duality to a nested set of three problems. One problem is zero degree of difficulty and the solution is obtained by solving a simple system of equations. The inclusion of a constraint restricting the force on the tool to be less than or equal to the breaking force provides a more realistic solution. This model is solved as a program with one degree of difficulty. Finally the behavior of the machining cost per part is studied parametrically as a function of axial depth.

Introduction to linear and convex programming

Abstract: Preface 1. Geometry and linear algebra 2. Linear programming 3. Elementary convex analysis 4. Non-linear programming Comments on exercises References Index.

Two parallel algorithms for the convex hull problem in a two dimensional space

Abstract: Two parallel algorithms for determining the convex hull of a set of data points in two dimensional space are presented. Both are suitable for MIMD parallel systems. The first is based on the strategy of divide-and-conquer, in which some simplest convex-hulls are generated first and then the final convex hull of all points is achieved by the processes of merging 2 sub-convex hulls. The second algorithm is by the process of picking up the points that are necessarily in the convex hull and discarding the points that are definitely not in the convex hull. Experimental results on a MIMD parallel system of 4 processors are analysed and presented.

Zur analytischen und algorithmischen behandlung eines geometrisehem optimierungsproblems von j. steiner

Abstract: This article deals with the following geometrical problem of optimization. Given a convex m –gon in the plane. Find the embedded convex domain which touches each side of and having the smallest per...

A greedy algorithm for solving a class of convex programming problems and its connection with polymatroid theory

Abstract: We present a greedy algorithm for solving a special class of convex programming problems and establish a connection with polymatroid theory which yields a theoretical explanation and verification of the algorithm via some recent results of S. Fujishige.

Definition and properties of a particular notion of convexity

Abstract: In an attempt to develop a general theory of nonlinear equations with exactly three solutions in general Hilbert spaces, we have noticed that a particular notion of convexity plays a key role. This paper is devoted to the study of some of its basic properties and we show how it generalizes homogeneous convex functionals on the one hand and some special convex functions of one real variable on the other hand. The important case of functionals associated with Nemytskii's operators is treated as an example.