Showing papers on "Convex optimization published in 1974"

A method of conjugate subgradients for minimizing nondifferentiable functions

Abstract: An algorithm is described for finding the minimum of any convex, not necessarily differentiable, function f of several variables. The algorithm yields a sequence of points tending to the solution of the problem, if any, requiring only the calculation of f and one subgradient of f at designated points. Its rate of convergence is estimated for convex and for differentiable convex functions. For the latter, it is an extension of the method of conjugate gradients and terminates for quadratic functions.

Statistical thermodynamics of convex molecule fluids

Abstract: A theoretical study of the statistical mechanical description of systems composed of non-spherical convex molecules is made. Thermodynamic functions of one-component and multicomponent systems of particles interacting via the pair potential of the Kihara core type are expressed by integrals over the minimum distance between two interacting convex bodies and three angles characterizing the convex body geometry. The approach is applied to the hard convex body system where the averaged contact correlation function is introduced. Exploiting ideas of the scaled particle theory the approximate expressions for the averaged correlation functions are given in terms of the geometric functionals of hard convex bodies.

Global Maximization of a Convex Function with Linear Inequality Constraints

Abstract: This paper presents an algorithm for the global maximization of a convex function subject to linear inequality constraints. It is computationally finite and is designed to converge rapidly on problems in which there are few local optima or the global optimum is significantly better than most of the other local optima.

On the cones of tangents with applications to mathematical programming

Abstract: In this study, we present a unifying framework for the cones of tangents to an arbitrary set and some of its applications. We highlight the significance of these cones and their polars both from the point of view of differentiability and subdifferentiability theory and the point of view of mathematical programming. This leads to a generalized definition of a subgradient which extends the well-known definition of a subgradient of a convex function to the nonconvex case. As an application, we develop necessary optimality conditions for a min-max problem and show that these conditions are also sufficient under moderate convexity assumptions.

Optimization of design tolerances using nonlinear programming

Abstract: A possible mathematical formulation of the practical problem of computer-aided design of electrical circuits (for example) and systems and engineering designs in general, subject to tolerances onk independent parameters, is proposed. An automated scheme is suggested, starting from arbitrary initial acceptable or unacceptable designs and culminating in designs which, under reasonable restrictions, are acceptable in the worst-case sense. It is proved, in particular, that, if the region of points in the parameter space for which designs are both feasible and acceptable satisfies a certain condition (less restrictive than convexity), then no more than 2k points, the vertices of the tolerance region, need to be considered during optimization.

Another Proof that Convex Functions are Locally Lipschitz

Abstract: (1974). Another Proof that Convex Functions are Locally Lipschitz. The American Mathematical Monthly: Vol. 81, No. 9, pp. 1014-1016.

Fenchel and Lagrange Duality are Equivalent

Abstract: Supported in part by the U.S. Army Research Office (Durham) under Contract No. DAHC04-73-C-0032.

A simplicial algorithm for the nonlinear complementarity problem

Abstract: A triangulation of the nonnegative orthant and a special labeling of the vertices lead to a combinatorial procedure for seeking solutions or approximate solutions to the nonlinear complementarity problem under coercive-like assumptions on the problem functions. Derivatives are not required. Convergence is proved, computational considerations are discussed, and some preliminary applications to convex programming and saddle point computation, along with numerical results, are presented.

Optimal filter design subject to output sidelobe constraints: Computational algorithm and numerical results

Abstract: An algorithm is presented for the design of optimal detection filters in radar and communications systems, subject to inequality constraints on the maximum output sidelobe levels. This problem was reduced in an earlier paper (Ref. 1) to an unconstrained one in the dual space of regular Borel measures, with a nondifferentiable cost functional. Here, the dual problem is solved via steepest descent, using the directional Gateaux differential. The algorithm is shown to be convergent, and numerical results are presented.

Necessary Conditions for Nonsmooth Variational Problems

Abstract: We describe in this article a scheme of necessary conditons for variational problems which are devoid of the customary smoothness and convexity assumptions. This is a descriptive paper based upon the author’s dissertation [1]. Proofs are omitted here; they will appear elsewhere.

Optimal filter design subject to output sidelobe constraints - Theoretical considerations

Abstract: The design of filters for detection and estimation in radar and communications systems is considered, with inequality constraints on the maximum output sidelobe levels. A constrained optimization problem in Hilbert space is formulated, incorporating the sidelobe constraints via a partial ordering of continuous functions. Generalized versions (in Hilbert space) of the Kuhn-Tucker and duality theorems allow the reduction of this problem to an unconstrained one in the dual space of regular Borel measures.

On the best representation of scattering data by analytic functions inL2-norm with positivity constraints

Abstract: The problem of representing scattering data given on the boundary of the analyticity domain by analytic functions satisfying unitarity is investigated. The optimal representation inL2-norm is shown to be the solution of a constrained convex optimization problem in some Hilbert space of analytic functions. A duality optimization theorem based on the generalized Lagrange multiplier technique is applied for solving this problem. The method is found to easily accommodate unequal errors on the real and imaginary parts of the amplitude and the case of data given along a limited part of the boundary.

The Transportation-Production Problem

Abstract: A transportation-production problem with increasing marginal production costs and linear shipping costs is considered. This problem is shown to be a convex programming problem, and an efficient iterative solution technique is presented. Numerical results are presented for various problems having 2,500 variables, 100 linear constraints, and 2,500 nonnegativity constraints. As expected, the number of iterations for an accurate solution depends on the degree of non-linearity of the objective function. Computing times on the CDC Cyber 70, Model 72, varied from 10 to 70 seconds among the different problems.

Sortie Allocation by a Nonlinear Programming Model for Determining a Munitions Mix

Abstract: : The report presents a mathematical programming approach to maximize a military objective function which is subject to resource constraints. The formulation takes account of the diminishing returns obtained with incremental capability increases, thus producing a problem of the convex programming type. A very general nonlinear programming algorithm is presented, along with a discussion of its numerical properties and a proof of its convergence for the convex programming problem. This is followed by a description of the computer input for the specific problem studied, and by an example problem with its associated computer output. An appendix details the use of the general algorithm for applications that differ from the one considered here. This usage involves altering several PL/1 procedures to the form desired.

Note on an extension of “Davidon” methods to nondifferentiable functions

Abstract: Some properties of “Davidon”, or variable metric, methods are studied from the viewpoint of convex analysis; they depend on the convexity of the function to be minimized rather than on its being approximately quadratic. An algorithm is presented which generalizes the variable metric method, and its convergence is shown for a large class of convex functions.

Application of duality theory to a class of composite cost control problems

Abstract: Let a control system be described by a continuous linear map ℒ* from the input spaceU* (some dual Banach space) into the output spaceX* (some finite-dimensional normed space). Within the class of control problems where the constraints and cost are expressed in terms of the norms on the input and output spaces, the following two have had extensive coverage: (i)minimum effort problem: find, from amongst all inputs which have corresponding outputs lying in some closed sphere inX* centered on some desired outputx d *, an output of minimum norm; and (ii)minimum deviation problem: find, from amongst all inputs lying in some closed sphere inU*, an input having corresponding output at a minimum distance fromx d *. However, thecomposite cost problem, where we seek to minimizeF(∥u*∥, ∥x d * −x*∥) over elements satisfyingx* = ℒ*u* (F a certain kind of convex functional), has not received the same attention. This paper presents results for the composite cost problem paralleling known results for the minimum effort and deviation problems. It is hoped that a gap in the literature is thereby filled. We show that (a) a solution exists, (b) the solution can be characterized in terms of some closed hyperplaneH inX, and (c)H can be computed as being an element on which some concave functional over closed hyperplanes inX achieves its maximum. The treatment allows of infinite-dimensional output spaces. We make extensive use of recently developed duality theory.

Production Correspondences and Convex Algebra

Abstract: The production technology is usually represented in the quantity space by production sets, or by production functions and correspondences. Shephard [4] showed that in many cases production functions can be obtained from cost functions of a given technology and Uzawa [6] argued the 1–1 correspondence between these production functions and cost functions. These results have been generalized to production correspondences by Shephard [5]. He also showed that there exists a dual relation between cost structures (resp. output revenue structures) in the price space and production-input structures (resp. production-output structures) in the quantity space.

Lagrange Multipliers for an N-Stage Model in Stochastic Convex Programming

Abstract: Many decision processes are of a sequential nature and make use of information which is revealed progressively through the observation, at various times, of random variables with known distributions A simple model with discrete stages is the following

Approximately convex average sums of unbounded sets

Abstract: In this note we show that the average sum of a large but finite number of unbounded and open sets is approximately convex if their "degree of nonconvexity" is bounded.

A Simplicial Algorithm for the Mixed Nonlinear Complementarity Problem with Application to Convex Programming.

Abstract: : Given a continuous mapping f(x) from (R sup N) to (R sup n) the authors consider a problem in which some components of f(x) are required to satisfy a complementarity condition and the other components are required to be zero. This problem includes the nonlinear complementarity problem, the problem of finding a zero of a system of nonlinear equations, and the problem of finding a Kuhn-Tucker point of a nonlinear program with both equality and inequality constraints. A simplicial approximation algorithm for this problem is given and finite termination conditions are established. These conditions provide previously unknown existence results. Application of the algorithm to convex programming is described and computational experience presented. (Author)

General Lagrange multiplier theorems

Abstract: A cone constraint is used to develop a general Lagrange multiplier theorem for normed linear spaces. Conditions for the payoff functional multiplier to be less than zero are given for Banach spaces. Sufficiency theorems involving Lagrange multipliers are developed for abstract programming problems. Generalizations of certain properties of convex functions will be used for optimization problems.

On the convergence of SUMT

Abstract: This paper investigates some convergence properties of Fiacco and McCormick's SUMT algorithm. A convex programming problem is given for which the SUMT algorithm generates a unique unconstrained minimizing trajectory having an infinite number of accumulation points, each a global minimizer to the original convex programming problem.

F-Convex Functions.

Abstract: : Let F be a family of functions: (R sup n) maps to R. A function: (R sup n) maps to R is called F-convex if it is supported, at each point, by some member of F. For particular choices of F one obtains the convex functions: (R sup n) maps to R and the generalized convex functions in the sense of Beckenbach. F-convex functions are characterized and studied, retaining some essential results of classical convexity.

An optimization model for labour limited scheduling under a round robin policy

Abstract: A labour limited scheduling problem for a single server serving N machines under a round robin policy is studied here The order in which the server visits the different machines is determined so as to maximize job flow rate We investigate a class of policies termed as {Si; 0} policy The existence of such a class of policy is shown The problem of minimization of average in-process inventory per unit of time or the minimization of mean flow times of jobs through the machines is formulated as a convex programming problem Various theorems are proved which give us the optimal solution in a closed form A sample problem is solved using the results developed

Dualitätstheorie für hyperboiisehe und stückweise lineare konvexe optimierungsprobleme

Abstract: Using a generalized LANGRANGE function for a linear hyperbelie programming problem the author of the present paper construets a dual problem, which becomes a piecewise linear convex programming problem. For such dual problems, propositions on duality are proved containing in particular all propostions of the duality theory of linear programming.

Complementarity problem and duality over convex cones

Abstract: Abstract The complementarity problem is defined and studied for cases where the constraints involve convex cones, thus extending the real and complex complementarity problems. Special cases of the problem are equivalent to dual, linear or quadratic, programs over polyhedral cones.