Nondifferential optimization via adaptive smoothing

The problem of minimizing a nondifferential functionx f(x) (subject, possibly, to nondifferential constraints) is considered. Conventional algorithms are employed for minimizing a differential approximationf off (subject to differentiable approximations ofg). The parameter is adaptively reduced in such a way as to ensure convergence to points satisfying necessary conditions of optimality for the original problem.This research was supported by the UK Science and Engineering Research Council, the National Science Foundation under Grant No. ECS-8121149, and the Joint Services Electronics Program, Contract No. F49620-79-C-0178.     

Thermal analysis study of the effect of molecular weight on constrained amorphous phase in poly(phenylene sulfide)

We report a thermal analysis study of the effect of molecular weight on the amorphous phase structure of poly(phenylene sulfide), PPS, crystallized at temperatures just above the glass transition temperature. Thermal properties of Fortron PPS, having viscosity average molecular weights of 30000 to 91000, were characterized using temperature modulated differential scanning calorimetry (MDSC). We find that while crystallinity varies little with molecular weight, the heat capacity increment at the glass transition decreases as molecular weight decreases. This leads to a smaller liquid-like amorphous phase, and a larger rigid amorphous fraction, in the lower molecular weight PPS. For all molecular weights, constrained fraction decreases as the scan rate decreases.This research is supported by the U. S. Army Research Office through contract DAAH04-96-1-0009. The authors thank Hoechst Celanese for providing different molecular weight Fortron samples and Dr. George Collins for providing sample information. The authors acknowledge the assistance of Elizabeth Oyebode and Leonardo Grimaldi with sample preparation and MDSC work.     

An RQP algorithm using a differentiable exact penalty function for inequality constrained problems

In this paper we propose a recursive quadratic programming algorithm for nonlinear programming problems with inequality constraints that uses as merit function a differentiable exact penalty function. The algorithm incorporates an automatic adjustment rule for the selection of the penalty parameter and makes use of an Armijo-type line search procedure that avoids the need to evaluate second order derivatives of the problem functions. We prove that the algorithm possesses global and superlinear convergence properties. Numerical results are reported.     

Generalizations of Slater's constraint qualification for infinite convex programs

In this paper we study constraint qualifications and duality results for infinite convex programs (P) = inf{f(x): g(x) – S, x C}, whereg = (g ₁,g ₂) andS = S ₁ ×S ₂,S _i are convex cones,i = 1, 2,C is a convex subset of a vector spaceX, andf andg _i are, respectively, convex andS _i -convex,i = 1, 2. In particular, we consider the special case whenS ₂ is in afinite dimensional space,g ₂ is affine andS ₂ is polyhedral. We show that a recently introduced simple constraint qualification, and the so-called quasi relative interior constraint qualification both extend to (P), from the special case thatg = g ₂ is affine andS = S ₂ is polyhedral in a finite dimensional space (the so-called partially finite program). This provides generalized Slater type conditions for (P) which are much weaker than the standard Slater condition. We exhibit the relationship between these two constraint qualifications and show how to replace the affine assumption ong ₂ and the finite dimensionality assumption onS ₂, by a local compactness assumption. We then introduce the notion of strong quasi relative interior to get parallel results for more general infinite dimensional programs without the local compactness assumption. Our basic tool reduces to guaranteeing the closure of the sum of two closed convex cones.     

Convexlike and concavelike conditions in alternative,minimax, and minimization theorems

Convexlike and concavelike conditions are of interest for extensions of the Von Neumann minimax theorem. Since the beginning of the 80's, these conditions also play a certain role in deriving generalized alternative theorems of the Gordan, Motzkin, and Farkas type and Lagrange multiplier results for constrained minimization problems.In this paper, we study various known convexlike conditions for vector-valued functions on a set and investigate convexlike and concavelike conditions for real-valued functions on a product setC×D, where we are mainly interested in the relationships between these conditions. At the end of the paper, we point out several conclusions from our results for the above-mentioned mathematical fields.The author is indebted to Dr. R. Reemtsen and Dr. V. Jeyakumar for their helpful comments during the preparations of this paper.     

Chip-Firing Games on Directed Graphs

We consider the following (solitary) game: each node of a directed graph contains a pile of chips. A move consists of selecting a node with at least as many chips as its outdegree, and sending one chip along each outgoing edge to its neighbors. We extend to directed graphs several results on the undirected version obtained earlier by the authors, P. Shor, and G. Tardos, and we discuss some new topics such as periodicity, reachability, and probabilistic aspects.Among the new results specifically concerning digraphs, we relate the length of the shortest period of an infinite game to the length of the longest terminating game, and also to the access time of random walks on the same graph. These questions involve a study of the Laplace operator for directed graphs. We show that for many graphs, in particular for undirected graphs, the problem whether a given position of the chips can be reached from the initial position is polynomial time solvable.Finally, we show how the basic properties of the probabilistic abacus can be derived from our results.     

Computation of efficient solutions of discretely distributed stochastic optimization problems

In engineering and economics often a certain vectorx of inputs or decisions must be chosen, subject to some constraints, such that the expected costs (or loss) arising from the deviation between the outputA() x of a stochastic linear systemxA()x and a desired stochastic target vectorb() are minimal. Hence, one has the following stochastic linear optimization problem minimizeF(x)=Eu(A()x b()) s.t.xD, (1) whereu is a convex loss function on ^m , (A(), b()) is a random (m,n + 1)-matrix, E denotes the expectation operator andD is a convex subset of ⁿ . Concrete problems of this type are e.g. stochastic linear programs with recourse, error minimization and optimal design problems, acid rain abatement methods, problems in scenario analysis and non-least square regression analysis.Solving (1), the loss functionu should be exactly known. However, in practice mostly there is some uncertainty in assigning appropriate penalty costs to the deviation between the outputA ()x and the targetb(). For finding in this situation solutions hedging against uncertainty a set of so-called efficient points of (1) is defined and a numerical procedure for determining these compromise solutions is derived. Several applications are discussed.     

Facility location in the presence of forbidden regions,I: Formulation and the case of Euclidean distance with one forbidden circle

A constrained form of the Weber problem is formulated in which no path is permitted to enter a prespecified forbidden region $\text{R}$ of the plane. Using the calculus of variations the shortest path between two points $\text{x}\text{,}\text{y}\text{?}\text{R}$ which does not intersect $\text{R}$ is determined. If $\text{d(}\text{x}\text{,}\text{y}\text{)}$ is unconstrained distance, we denote the shortes distance along a feasible path by $\text{d}\text{(}\text{x}\text{y}\text{)}$. The constrained Weber problem is, then: given points $\text{x}{}_{\mathrm{j}}\text{?}\text{R}$ and positive weights w_j, j = 1,2,…,n, find a point $\text{x}\text{?}\text{R}$ such that

$\text{f(}\text{x}\text{)=}\text{Σ}\text{n}\text{j=1}\text{d(}\text{x}\text{,}\text{x}{}_{\mathrm{j}}\text{)}$

is a minimum.An algorithm is formulated for the solution of this problem when $\text{d(}\text{x}\text{,}\text{y}\text{)}$ is Euclidean distance and $\text{R}$ is a single circular region. Numerical results are presented.     

Stability theorems for infinitely constrained mathematical programs

The primary concern of this paper is to investigate stability conditions for the mathematical program: findx E ⁿ that maximizesf(x):g ^j(x)0 for somej J, wheref is a real scalarvalued function and eachg is a real vector-valued function of possibly infinite dimension. It should be noted that we allow, possibly infinitely many, disjunctive forms. In an earlier work, Evans and Gould established stability theorems wheng is a continuous finite-dimensional real-vector function andJ=1. It is pointed out that the results of this paper reduce to the Evans-Gould results under their assumptions. Furthermore, since we use a slightly more general definition of lower and upper semicontinuous point-to-set mappings, we can dispense with the continuity ofg (except in a few instances where it is implied by convexity assumptions).     

Generalized Levitin-Polyak Well-Posedness for Generalized Semi-Infinite Programs

In this article, we study the generalized Levitin-Polyak well-posedness of generalized semi-infinite programs. We first give the criteria and characterizations for two types of well-posedness. We then establish the convergence of a class of penalty methods under the assumption of generalized type I Levitin-Polyak well-posedness.