Showing papers on "Convex optimization published in 1969"

On the theory of semi‐infinite programming and a generalization of the kuhn‐tucker saddle point theorem for arbitrary convex functions

Abstract: We first present a survey on the theory of semi-infinite programming as a generalization of linear programming and convex duality theory. By the pairing of a finite dimensional vector space over an arbitrarily ordered field with a generalized finite sequence space, the major theorems of linear programming are generalized. When applied to Euclidean spaces, semi-infinite programming theory yields a dual theorem associating as dual problems minimization of an arbitrary convex function over an arbitrary convex set in n-space with maximization of a linear function in non-negative variables of a generalized finite sequence space subject to a finite system of linear equations. We then present a new generalization of the Kuhn-Tucker saddle-point equivalence theorem for arbitrary convex functions in n-space where differentiability is no longer assumed.

Locating facilities in three-dimensional space by convex programming

Abstract: We consider the problem of simultaneously locating any number of facilities in three-dimensional Euclidean space. The criterion to be satisfied is that of minimizing the total cost of some activity between the facilities to be located and any number of fixed locations. Any amount of activity may be present between any pair of the facilities themselves. The total cost is assumed to be a linear function of the inter-facility and facility-to-fixed locations distances. Since the total cost function for this problem is convex, a unique optimal solution exists. Certain discontinuities are shown to exist in the derivatives of the total cost function which previously has prevented the successful use of gradient computing methods for locating optimal solutions. This article demonstrates the use of a created function which possesses all the necessary properties for ensuring the convergence of first order gradient techniques and is itself uniformly convergent to the actual objective function. Use of the fitted function and the dual problem in the case of constrained problems enables solutions to be determined within any predetermined degree of accuracy. Some computation results are given for both constrained and unconstrained problems.

Separation and support properties of convex sets: a survey.

Abstract: : Separation and support properties of convex sets are widely recognized as natural and important tools in the theory of optimal control. The present report provides a comprehensive survey of such properties, going far beyond those that have actually been applied in control theory up to the present time. The discussion is entirely in the neutral language of convexity theory and thus should be useful to workers in the many other areas (in addition to control theory) in which separation and support properties have been applied. (Author)

Convex Functions and Duality in Optimization Problems and Dynamics

Abstract: Everyone is aware of the importance of convex sets in the study of optimization problems. Much of the modern theory of convex functions is less well known, however, and for this reason has not sufficiently been exploited. This is true especially of Fenchel’s theory of conjugacy [11], which ought to be made the vehicle for all results involving duality. Fenchel’s theory and some of its consequences will be described below.

A Stochastic Multiperiod Multimode Transportation Model

Abstract: This paper develops a dynamic programming model for selecting an optimal combination of transportation modes over a midtiperiod planning horizon. The formulation explicitly incorporates uncertainty regarding future requirements or demands for a number of commodity classes. In addition to determining the optimal modes to employ, the model assigns individual commodity classes to various modes, determines which supply points serve which destinations, and reroutes carriers from destinations to alternative sources where they will be most effective. The model is formulated as an optimal discrete time stochastic control problem where cost is quadratic and dynamic equations linear in the state and control variables. This model may be solved in closed form by an efficient dynamic programming algorithm that permits the treatment of relatively large scale systems. Also developed is an alternative, generally suboptimal method of solution, based upon solving a sequence of convex programming problems over time. This te...

On the order of approximation of convex functions by rational functions

Abstract: We show that for arbitrary convex functions the order of approximation (in the metric C[a, b]) by rational functions of degree no higher than n does not exceed the quantity CMln²n/n (C an absolute constant, M the maximum modulus of the convex function). We prove also the existence of a convex function whose order of approximation is greater than 1/n ln²n.

An Extremum Property of Convex Functions

Abstract: (1969). An Extremum Property of Convex Functions. The American Mathematical Monthly: Vol. 76, No. 8, pp. 921-922.

Optimal Control of Linear Distributed Parameter Systems

Abstract: Publisher Summary This chapter discusses optimal control of linear distributed parameter systems. The preponderance of results in optimal control theory pertains to systems governed either by ordinary differential equations or difference equations. The chapter presents a view of optimal control problems for distributed parameter systems that has evolved from studying their properties with the techniques of functional analysis. This approach has been successful in uncovering new results about the theory of such problems. The central ideas of optimal control theory are retained in this development for distributed parameter systems. The chapter presents a basic approach to induce a parameterization into the class of allowable controls such that the optimization problem may be solved to within any required accuracy by an approximate control described by a finite number of parameters. The appropriate values for these parameters may then be found by convex programming.

Monoextremal Representations of a Class of Minimax Problems

Abstract: It is shown that minimax problems with convex-concave payoff and convex inequality constraints can be reduced to solving an equivalent constrained minimization problem. The objective function of the minimization problem is not convex, nor is the set of feasible solutions convex. Nevertheless, it is shown that any point which satisfies the first order necessary conditions for optimality must be a global minimum.

A simple proof of a general duality theorem of convex programming

Abstract: The first part of this note presents concisely and partially proves in logical terms the relations betweenUzawa's andKuhn andTucker's equivalence theorems of nonlinear programming. In the second part we give simple mathematical proofs of two lemmata linking theKuhn-Tucker conditions and dual solutions and use them to establish a nonlinear duality theorem of considerable generality.

Duality for non-linear programming in aBanach space

Abstract: A constrained maximization problem in a realBanach space is considered, where the objective function is a non-linear pseudo concave functional and constraints are given bym non linear quasi convex functionals. The optimality conditions and converse duality theorem, given byRitter, have been extended to this class of programming problems.

Every norm is not logarithmically convex

Abstract: This correspondence relates to the remark in a recent paper by D.G. Luenberger [ibid., vol. SSC-4, pp. 182-188, July 1968] that any norm defined on a vector space is a real convex function. Although this is a well-known fact in mathematics, a less well-known fact is that every logarithmically convex function is positive and convex, but not conversely, i.e., there are positive convex functions which are not logarithmically convex. As the above title indicates, norms are such functions. This mathematical remark relates to systems science through several areas of application where logarithmic convexity is a highly useful property. In particular, Klinger and Mangasarian ["Logarithmic convexity and geometric programming," J. Math. Anal. and Appl., vol. 24, pp. 388-408, November 1968] mention optimization of multiplicative criteria, reliability theory, and electrical network synthesis, and examine geometric programming in detail.

On a convex function useful in determination of linear discriminant functions

Abstract: The problem of determining a linear discriminant function to minimize probability of error in distinguishing between two measurement classes may, as is well known, often be approximated by a problem of minimizing some convex function. Many particular convex functions have been proposed for this purpose. While the relations between the real problem and the approximation are of critical importance, such relations are frequently unclear or even unsatisfactory. Here we examine a particular choice and show that it has a collection of desirable attributes, relating to the real problem, which are not known to be possessed by any other choice.