Showing papers on "Convex optimization published in 1977"

An Extension of Duality-Stability Relations to Nonconvex Optimization Problems

Abstract: By an effective extension of the conjugate function concept a general framework for duality-stability relations in nonconvex optimization problems can be studied. The results obtained show strong correspondences with the duality theory for convex minimization problems. In specializations to mathematical programming problems the canonical Lagrangian of the model appears as the extended Lagrangian considered in exterior penalty function methods.

Multivalued convexity and optimization: A unified approach to inequality and equality constraints

Abstract: Multivalued functions satisfying a general convexity condition are examined in the first section. The second section establishes a general transposition theorem for such functions and develops an abstract multiplier principle for them. In particular both convex inequality and linear equality constraints are seen to satisfy the same generalized constraint qualification. The final section examines quasi-convex programmes.

On generalized means and generalized convex functions

Abstract: Properties of generalized convex functions, defined in terms of the generalized means introduced by Hardy, Littlewood, and Polya, are easily obtained by showing that generalized means and generalized convex functions are in fact ordinary arithmetic means and ordinary convex functions, respectively, defined on linear spaces with suitably chosen operations of addition and multiplication. The results are applied to some problems in statistical decision theory.

Generalizations of Farkas’ Theorem

Abstract: A unified treatment is given of generalizations of Farkas’ theorem on linear inequalities to arbitrary convex cones and to dual pairs of real vector spaces of arbitrary dimension. Various theorems for locally convex spaces readily follow. The results are applied to duality and converse duality theory for linear programming and to a generalization of the Kuhn–Tucker theorem, both of these in spaces of arbitrary dimension and with inequalities involving arbitrary convex cones.

Qualitative analysis of basic notions in parametric convex programming. II. Parameters in the objective function

Abstract: The paper presents a qualitative analysis of basic notions in parametric convex programming for convex programs with parameters in the objective function. These notions are the set of feasible parameters, the solvability set and the stability sets of the first and of the second kind. The functions encountered in the paper are assumed to possess first order partial continuous derivatives on $R^n$, the parameters assume arbitrary nonnegative real values and therefore the results obtained in the paper can be used for a wide class of convex programs.

Minimax procedure for a class of linear programs under uncertainty.

Abstract: tribution functions are otherwise unspecified. A minimax solution of the stochastic programming model is obtained by solving an equivalent deterministic convex programming problem. We derive these deterministic equivalents under different assumptions regarding the stochastic nature of the random parameters. IN FORMULATING a stochastic linear programming model, we generally assume definite probability distribution for the parameters (A, b, c) of the model. In this note we avoid making the assumption that the precise form of the probability distribution of the parameters is known. What we assume, however, is that the random aij and bi elements have known (finite) means and variances. The problem is then to obtain a minimax solution that minimizes the maximum of the objective function over all distributions with the given mean and standard deviation. The situation of a decision maker facing an unknown probability distribution can be viewed as a zero-sum game against nature. Zackova [3] proves that the general min-max theorem holds in this case if the set P of all possible distributions is assumed to be convex and compact (in the sense of Levy's distance). However, neither the above result nor its proof presents any effective method for finding the value of the game or determining explicit solutions. In Section 1 we obtain some results that are used later in determining minimax solutions under different assumptions regarding the stochastic nature of the random parameters of a linear programming model. Section 2 considers a stochastic linear programming problem with random RHS elements and obtains a minimax solution of the problem as an optimal solution of an equivalent deterministic convex separable programming problem. Section 3 presents similar results for the case of random aii elements.

Technical Note-Minimax Procedure for a Class of Linear Programs under Uncertainty

Abstract: We consider a linear programming problem with random aij and bi elements that have known finite mean and variance, but whose distribution functions are otherwise unspecified. A minimax solution of the stochastic programming model is obtained by solving an equivalent deterministic convex programming problem. We derive these deterministic equivalents under different assumptions regarding the stochastic nature of the random parameters.

A Discrete-Convex Programming Approach to the Simultaneous Optimization of Land Use and Transportation

Abstract: Three design problems are discussed in this article. First, it is shown that the network design problem with congestion reduces to an all-or nothing traffic assignment problem under some assumptions on the congestion function and the investment cost function. Second, the land use design problem is formulated as an extension of the Koopmans-Beckmann problem and a heuristic is proposed to solve this problem. Third, it is shown that the seemingly more complex problem of designing jointly a land-use plan and a transportation network reduces to a pure land-use design problem. All that is needed to solve the joint optimization problem is a shortest path algorithm and a heuristic to solve the land use design problem. Computational experience is reported for each algorithm.

Qualitative analysis of basic notions in parametric convex programming. I. Parameters in the constraints

Abstract: The paper presents a qualitative analysis of basic notions in parametric convex programming for convex programs with parameters in the righthand sides of the constraints. These notions are the set of feasible parameters, the solvability set and the stability sets of the first and of the second kind. The functions encountered in the paper are assumed to possess first order partial continuous derivatives on $R^n$, the parameters assume arbitrary real values and therefore the results obtained in the paper can be used for a wide class of convex programs.

Stochastic transportation problems and other newtork related convex problems

Abstract: A class of convex programming problems with network type constraints is addressed and an algorithm for obtaining the optimal solution is described. The stochastic transportation problem (minimize shipping costs plus expected holding and shortage costs at demand points subject to limitations on supply) is shown to be amenable to the solution technique presented. Network problems whose objective function is non-separable and network problems with side constraints are also shown to be solvable by the algorithm. Several large stochastic transportation problems with up to 15,000 variables and non-negativity constraints and 50 supply constraints are solved.

Necessary and Sufficient Conditions for a Pareto Optimum in Convex Programming

Abstract: : Necessary and sufficient conditions for Pareto optimality are derived for problems involving convex criteria and convex constraints. For a wide class of convex functions, the characterization of Pareto optimality is given in terms of systems of linear programs, which, under suitable regularization conditions, reduce to a single linear program. The consideration of a system of linear programs and their duals leads naturally to a system of partial prices associated with a Pareto optimum. (Author)

On the Solution of Convex Knapsack Problems with Bounded Variables.

Abstract: : In this paper, a recursive method is presented to solve separable differentiable convex knapsack problems with bounded variables. The method differs from classical optimization algorithms of convex programming and determines at each iteration the optimal value of at least one variable. Applications of such problems are frequent in resource allocation and recently have shown to be useful in hierarchical production planning. Computational results are presented.

A Dual Optimization Framework for Some Problems of Information Theory and Statistics.

Abstract: : A new dual optimization framework for some problems of information theory and statistics is developed in the form of dual convex programming problems and their duality theory. It extends the work for finite discrete distributions to the case of general measures. Although the primal problem (constrained relative entropy) is an infinite dimensional one, the dual problem is a finite dimensional one without constraints and involving only exponential and linear terms. Applications range from mathematical statistics and statistical mechanics to traffic engineering, marketing and economics.

Second-order characterizations of pseudoconvex quadratic functions

Abstract: The paper deals with strictly pseudoconvex quadratic functions on open convex sets. It presents second-order conditions in terms of an extended Hessian and in terms of bordered determinants. All of the conditions are both necessary and sufficient.

Global stability of optimization problems

Abstract: This paper presents necessary and sufficient conditions for the stability of the problem . Here M is a subset of a metric space X, λ is an element of some set ⋀ “with convergence” and f is a functional defined on the Cartesian product X×⋀. These conditions apply to the upper and lower semiconformity of the function and the upper semiconformity of the point-to-set mapping . The used set-convergence is less strong than the convergence induced by the Hausdorff metric. As conclusions theorems on the relationship between the f 0 and [fcirc] upper semiconformity and sufficient stability-conditions for some general problems (especially quasi convex programming) are received. The necessity of certain suppositions is illustrated by appropriate examples.

Convex Programming and the Lexicographic Multicriteria Problem.

Abstract: Mathematical programming formulation of the convex lexicographic multi-criteria problems typically lacks a constraint qualification. Therefore the classical Kuhn-tucker theory fails to characterize their optimal solutions. Furthermore, numerical methods for solving the lexicographic problems are virtually nonexistent. This paper shows that using a recent theory of convex programming, which is free of a constraint qualification assumption, it is possible both to characterize and to calculate the optimal solutions of the convex lexicographic problem.

How Can Specialized Discrete and Convex Optimization Methods Be Married

Abstract: Numerous practical problems involve both logical design choices and continuous-valued decision variables which are predicated in some manner on the logical design. For instance: industrial scheduling problems usually involve both sequencing and the determination of how continuously divisible resources should be applied for the chosen sequence, and network synthesis problems involve both the logical design of the network and the programming of flows for the chosen design. Many such problems which are difficult to solve directly as a whole have the tantalizing properties that (a) specialized algorithms (discrete or combinatorial) are available for close relatives of the logical design aspect of the problem, and (b) for any particular logical design the resulting continuous optimization problem can be solved by an available convex programming method (usually by LP or a network flow technique). This raises the question of how the two specialized types of algorithms can be married to provide an effective overall approach to the problem. Several possible kinds of marriages are surveyed and attractive opportunities for further research are pointed out.

Sensitivity for non Convex Optimization Problems

Abstract: Consider a mathematical optimization problem which is depending on a parameter m ∈ M max f(x, m) (pm) Open image in new window .

On optimizing certain nonlinear convex functions which are partially defined by a simulation process

Abstract: This paper considers the problem of optimizing a nonlinear convex function which is, in part, defined by a simulation process. An iterative technique, based on contractive mapping properties, is given by which a selected class of such functions can be optimized.

Technical Note-Column Dropping in the Dantzig-Wolfe Convex Programming Algorithm: Computational Experience

Abstract: This paper presents some computational experience on column dropping using the Dantzig-Wolfe convex programming algorithm. Computational results show virtually no difference in the measures of computational efficiency when all columns are retained and when only basic columns are retained.

Convex programs having some linear constraints

Abstract: The problem of concern is the minimization of a convex function over a normed space (such as a Hilbert space) subject to the constraints that a number of other convex functions are not positive. As is well known, there is a dual maximization problem involving Lagrange multipliers. Some of the constraint functions are linear, and so the Uzawa, Stoer, and Witzgall form of the Slater constraint qualifications is appropriate. A short elementary proof is given that the infimum of the first problem is equal to the supremum of the second problem.

Asymptotic area and perimeter of sums of random plane convex sets

Abstract: : The asymptotic distribution of functionals associated with normed sums of random plane convex sets is considered. Methods involving symmetric statistics and weak convergence of stochastic processes are used to examine area and perimeter in particular.

The Exact Order of the best Approximation to Convex Functions by Rational Functions

Abstract: We show that the least uniform rational deviations from the function , continuous and convex on the interval , satisfy the condition as , and that uniformly for the continuous convex functions whose absolute values are bounded by unity. These estimates are precise with respect to the rate of decrease of the right-hand sides.Bibliography: 16 titles.