Showing papers on "Convex optimization published in 1984"

Lipschitz Behavior of Solutions to Convex Minimization Problems

Abstract: We derive the Lipschitz dependence of the set of solutions of a convex minimization problem and its Lagrange multipliers upon the natural parameters from an inverse function theorem for set-valued maps. This requires the use of contingent and Clarke derivatives of set-valued maps, as well as generalized second derivatives of convex functions.

An Efficient Method for Computing Traffic Equilibria in Networks with Asymmetric Transportation Costs

Abstract: In the presence of several user categories or transportation modes, or when transportation costs on each arc of a network depend on the flows on adjacent arcs, the traffic equilibrium problem may be expressed as a variational problem. Methods for determining traffic equilibria are then adaptations of techniques for solving variational inequalities. In this paper, we present a new convex optimization formulation for the general traffic equilibrium problem, and propose a simple iterative method for calculating traffic equilibria, which essentially involves postoptimizing a linear sub-problem at each iteration. Preliminary computational results are reported.

Convex partitions of polyhedra: a lower bound and worst-case optimal algorithm

Abstract: The problem of partitioning a polyhedron into a minimum number of convex pieces is known to be NP-hard. We establish here a quadratic lower bound on the complexity of this problem, and we describe an algorithm that produces a number of convex parts within a constant factor of optimal in the worst case. The algorithm is linear in the size of the polyhedron and cubic in the number of reflex angles. Since in most applications areas, the former quantity greatly exceeds the latter, the algorithm is viable in practice.

Totally balanced games arising from controlled programming problems

Abstract: A cooperative game in characteristic-function form is obtained by allowing a number of individuals to esercise partial control over the constraints of a (generally nonlinear) mathematical programming problem, either directly or through committee voting. Conditions are imposed on the functions defining the programming problem and the control system which suffice to make the game totally balanced. This assures a nonempty core and hence a stable allocation of the full value of the programming problem among the controlling palyers. In the linear case the core is closely related to the solutions of the dual problem. Applications are made to a variety of economic models, including the transferable utility trading economies of Shapley and Shubik and a multishipper one-commodity transshipment model with convex cost functions and concave revenue functions. Dropping the assumption of transferable utility leads to a class of controlled multiobjective or ‘Pareto programming’ problems, which again yield totally balanced games.

Directional differentiability of the optimal value function in a nonlinear programming problem

Abstract: A parameterized nonlinear programming problem is considered in which the objective and constraint functions are twice continuously differentiable. Under the assumption that certain multiplier vectors appearing in generalized second-order necessary conditions for local optimality actually satisfy the weak sufficient condition for local optimality based on the augmented Lagrangian, it is shown that the optimal value in the problem, as a function of the parameters, is directionally differentiable. The directional derivatives are expressed by a minimax formula which generalizes the one of Gol’shtein in convex programming.

Equivalence of some quadratic programming algorithms

Abstract: We formulate a general algorithm for the solution of a convex (but not strictly convex) quadratic programming problem. Conditions are given under which the iterates of the algorithm are uniquely determined. The quadratic programming algorithms of Fletcher, Gill and Murray, Best and Ritter, and van de Panne and Whinston/Dantzig are shown to be special cases and consequently are equivalent in the sense that they construct identical sequences of points. The various methods are shown to differ only in the manner in which they solve the linear equations expressing the Kuhn-Tucker system for the associated equality constrained subproblems. Equivalence results have been established by Goldfarb and Djang for the positive definite Hessian case. Our analysis extends these results to the positive semi-definite case.

Exact penalty functions and stability in locally Lipschitz programming

Abstract: In this paper we extend the theory of exact penalty functions for nonlinear programs whose objective functions and equality and inequality constraints are locally Lipschitz; arbitrary simple constraints are also allowed. Assuming a weak stability condition, we show that for all sufficiently large penalty parameter values an isolated local minimum of the nonlinear program is also an isolated local minimum of the exact penalty function. A tight lower bound on the parameter value is provided when certain first order sufficiency conditions are satisfied. We apply these results to unify and extend some results for convex programming. Since several effective algorithms for solving nonlinear programs with differentiable functions rely on exact penalty functions, our results provide a framework for extending these algorithms to problems with locally Lipschitz functions.

A Minimax Theorem

Abstract: A new, general criterion is given for ensuring that a closed saddle function has a nonempty compact set of saddlepoints. Under this criterion it is shown also that every minimaximizing sequence clusters around some saddlepoint. A comparable theorem is given for semicontinuous quasi-saddle functions. The new criterion is applied to constrained saddlepoint problems and to the Fenchel-Rockafellar duality model for constrained convex minimization. Finally, the relationship to existing saddlepoint results is explored in detail.

Convex programming formulations of the asymmetric traffic assignment problem

Abstract: Recently introduced optimization formulations of the asymmetric traffic assignment problem are developed. The duality of two formulations is shown, and conditions for convexity and differentiability of the objective functions are given. Convexity conditions are also given for the family of formulations recently introduced by Smith (1983). When the travel cost vector is affine and monotone, all of the formulations are shown to be convex programming problems. Algorithmic implications of the results are discussed.

On the existence of convex decompositions of partially separable functions

Abstract: The concept of a partially separable functionf developed in [4] is generalized to include all functionsf that can be expressed as a finite sum of element functionsfi whose Hessians have nontrivial nullspacesNi, Such functions can be efficiently minimized by the partitioned variable metric methods described in [5], provided that each element functionfi is convex. If this condition is not satisfied, we attempt toconvexify the given decomposition by shifting quadratic terms among the originalfi such that the resulting modified element functions are at least locally convex. To avoid tests on the numerical value of the Hessian, we study the totally convex case where all locally convexf with the separability structureNi1 have a convex decomposition. It is shown that total convexity only depends on the associated linear conditions on the Hessian matrix. In the sparse case, when eachNi is spanned by Cartesian basis vectors, it is shown that a sparsity pattern corresponds to a totally convex structure if and only if it allows a (permuted) LDLT factorization without fill-in.

On stability analysis in mathematical programming

Abstract: Sufficient conditions for upper semicontinuity of approximate solutions and continuity of the values of mathematical programming problems with respect to data perturbations are obtained by using the variational convergence, thereby generalizing many known results. Upper and approximate lower semicontinuity of solutions and multipliers in infinite dimensional convex programming are obtained under gamma convergence of the data.

A weighted gram-schmidt method for convex quadratic programming*

Abstract: Range-space methods for convex quadratic programming improve in efficiency as the number of constraints active at the solution decreases. In this paper we describe a range-space method based upon updating a weighted Gram-Schmidt factorization of the constraints in the active set. The updating methods described are applicable to both primal and dual quadratic programming algorithms that use an active-set strategy.

Iterative Methods for the Convex Feasibility Problem

Abstract: The problem of finding a point in the intersection of a finite family of closed convex sets in the Euclidean space is considered here. Several iterative methods for its solution are reviewed and some connections between them are pointed out.

Conjugacy in quasi-convex programming

Abstract: This paper develops a symmetric conjugate relation for quasi-convex functions The concept of an evenly quasi-convex function is introduced and it is shown that this is the required property for a duality framework in quasi-convex programming

An algorithm for solving two-level convex optimization problems

Abstract: This paper provides an algorithm for solving two-level convex optimization problems. The algorithm is based on the subgradient formula for the upper level objective function which is not generally ...

Solution of linear and nonlinear algebraic problems with sharp, guaranteed bounds

Abstract: In this paper new methods for solving algebraic problems with high accuracy are described. They deliver bounds for the solution of the given problem with an automatic verification of the correctness. Examples of such problems are systems of linear equations, over- and underdetermined systems of linear equations, algebraic eigenvalue problems, nonlinear systems, polynomial zeros, evaluation of arithmetic expressions, linear, quadratic and convex programming and others. The new methods apply for these problems over the space of real numbers, complex numbers as well as real intervals and complex intervals.

Generalized equations: Solvability and regularity

Abstract: Generalized equations defined by an upper semicontinuous, convex-valued multifunction are considered. Based on some necessary and sufficient condition for such equation to be solvable a certain type of regularity of a generalized equation is studied. The application of the presented conditions for regularity leads to an implicit function theorem which uses a general notion of a derivative of a multifunction. Further, quantitative stability of the (subgradient) Kuhn-Tuckerpoints to convex optimization problems is examined from the point of view how to reduce the case of general continuous perturbations to the one of linearly perturbed only.

On differentiability with respect to parameter of solutions to convex optimal control problems subject to state space constraints

Abstract: A family of convex optimal control problems that depend on a real parameterh is considered. The optimal control problems are subject to state space constraints. It is shown that under some regularity conditions on data the solutions of these problems as well as the associated Lagrange multipliers are directionally-differentiable functions of the parameter. The respective right-derivatives are given as the solution and respective Lagrange multipliers for an auxiliary quadratic optimal control problem subject to linear state space constraints. If a condition of strict complementarity type holds, then directional derivatives become continuous ones.

A Finite Method for the Solution of a Multi-Resource Allocation Problem with Concave Return Functions

Abstract: It has recently been recognized that the problem of allocating continuous resources to several activities, each with a concave return function, has some elements in common with the Transportation Problem. This has inspired a search for solution algorithms for the allocation problem in which the approach used is similar to that applied to the Transportation Problem. This paper describes such an algorithm. The algorithm terminates and produces the exact optimal solution. A FORTRAN implementation of the algorithm solves a 12 resources/100 activities problem in less than 10 central processor seconds on a Cyber 173 computer.

Integration with respect to operator-valued measures with applications to quantum estimation theory

Abstract: This paper is concerned with the development of an integration theory with respect to operator-valued measures which is required in the study of certain convex optimization problems. These convex optimization problems in their turn are rigorous formulations of detection theory in a quantum communication context, which generalise classical (Bayesian) detection theory. The integration theory which is developed in this paper is used in conjunction with convex analysis in Banach spaces to give necessary and sufficient conditions of optimality for this class of convex optimization problems.

Subgradients of convex operators

Abstract: We give two general conditions for a convex operator taking values in an ordered vector space to possess subgradient operators. Various consequences are described an extension to abelian groups is given.

Exchange Algorithms, Error Estimations and Strong Unicity in Convex Programming and Chebyshev Approximation

Abstract: In the first section an exchange algorithm is given which is a generalization of the Remez algorithm: Using an idea of Topfer we consider at each step a finite sequence of finite sub-problems, which is called an optimization problem with respect to a chain of references. Modifying a strategy of Carasso and Laurent we replace in the.exchange theorem the zero-checks by practical δ-checks with δ > 0. This allows us to reduce the numerical calculations of the functional to be minimized, to raise the stability of the algorithm and to introduce multiple exchange techniques known from the Remez algorithm for getting faster convergence.

Nonlinear alternative theorems and nondifferentiable programming

Abstract: This paper provides some theorems of the alternative for non-linear functions (sublinearconvex) between topological vector spaces. These results are then applied to establish optimality conditions for convex programming and general non-differentiable programming problems.