The application of the principle of minimum cross-entropy to the characterization of the exponential-type probability distributions

Summary Systematic and simple characterizations are presented for several familiar distributions in exponential family by means of the principle of minimum cross-entropy (minimum discrimination information). The suitable prior distributions and the appropriate constraints on expected values are given for the underlying distributions.     

Controlled perturbations for quadratically constrained quadratic programs

Consider a minimization problem of a convex quadratic function of several variables over a set of inequality constraints of the same type of function. The duel program is a maximization problem with a concave objective function and a set of constrains that are essentially linear. However, the objective function is not differentiable over the constraint region. In this paper, we study a general theory of dual perturbations and derive a fundamental relationship between a perturbed dual program and the original problem. Based on this relationship, we establish a perturbation theory to display that a well-controlled perturbation on the dual program can overcome the nondifferentiability issue and generate an ε-optimal dual solution for an arbitrarily small number ε. A simple linear program is then constructed to make an easy conversion from the dual solution to a corresponding ε-optimal primal solution. Moreover, a numerical example is included to illustrate the potential of this controlled perturbation scheme.     

Deriving an unconstrained convex program for linear programming

Consider a linear programming problem in Karmarkar's standard form. By perturbing its linear objective function with an entropic barrier function and applying generalized geometric programming theory to it, Fang recently proposed an unconstrained convex programming approach to finding an epsilon-optimal solution. In this paper, we show that Fang's derivation of an unconstrained convex dual program can be greatly simplified by using only one simple geometric inequality. In addition, a system of nonlinear equations, which leads to a pair of primal and dual epsilon-optimal solutions, is proposed for further investigation.This work was partially supported by the North Carolina Supercomputing Center and a 1990 Cray Research Grant. The authors are indebted to Professors E. L. Peterson and R. Saigal for stimulating discussions.     

Sensitivity analysis in posynomial geometric programming

Sensitivity analysis results for general parametric posynomial geometric programs are obtained by utilizing recent results from nonlinear programming. Duality theory of geometric programming is exploited to relate the sensitivity results derived for primal and dual geometric programs. The computational aspects of sensitivity calculations are also considered.This work was part of the doctoral dissertation completed in the Department of Operations Research, George Washington University, Washington, DC. The author would like to express his gratitude to the thesis advisor, Prof. A. V. Fiacco, for overall guidance and stimulating discussions which inspired the development of this research work.     

A canonical dual approach for solving linearly constrained quadratic programs

This paper provides a canonical dual approach for minimizing a general quadratic function over a set of linear constraints. We first perturb the feasible domain by a quadratic constraint, and then solve a “restricted” canonical dual program of the perturbed problem at each iteration to generate a sequence of feasible solutions of the original problem. The generated sequence is proven to be convergent to a Karush-Kuhn-Tucker point with a strictly decreasing objective value. Some numerical results are provided to illustrate the proposed approach.     

Generalized geometric programming applied to problems of optimal control: I. Theory

The interest in convexity in optimal control and the calculus of variations has gone through a revival in the past decade. In this paper, we extend the theory of generalized geometric programming to infinite dimensions in order to derive a dual problem for the convex optimal control problem. This approach transfers explicit constraints in the primal problem to the dual objective functional.The authors are indebted to the referees for suggestions leading to improvement of the paper.     

Generalized geometric programming applied to problems of optimal control: Part I,theory

A modification to the formulation in Ref. 1 is given.     

Transcendental geometric programs

This paper treats a class of posynomial-like functions whose variables may appear also as exponents or in logarithms. It is shown that the resulting programs, called transcendental geometric programs, retain many useful properties of ordinary geometric programs, although the new class of problems need not have unique minima and cannot, in general, be transformed into convex programs. A duality theory, analogous to geometric programming duality, is formulated under somewhat more restrictive conditions. The dual constraints are not all linear, but the notion ofdegrees of difficulty is maintained in its geometric programming sense. One formulation of the dual program is shown to be a generalization of the chemical equilibrium problem where correction factors are added to account for nonideality. Some of the computational difficulties in solving transcendental programs are discussed briefly.This research was partially supported by the National Institute of Health Grant No. GM-14789; Office of Naval Research under Contract No. N00014-75-C-0276; National Science Foundation Grant No. MPS-71-03341 A03; and the US Atomic Energy Commission Contract No. AT(04-3)-326 PA #18.     

A Benders approach for the constrained minimum break problem

This paper presents a hybrid IP/CP algorithm for designing a double round robin schedule with a minimal number of breaks. Both mirrored and non-mirrored schedules with and without place constraints are considered. The algorithm uses Benders cuts to obtain feasible home-away pattern sets in few iterations and this approach leads to significant reductions in computation time for hard instances. Furthermore, the algorithm is capable of solving a number of previously unsolved benchmark problems for the Traveling Tournament Problem with constant distances.     

A strong duality theorem for the minimum of a family of convex programs

We produce a duality theorem for the minimum of an arbitrary family of convex programs. This duality theorem provides a single concave dual maximization and generalizes recent work in linear disjunctive programming. Homogeneous and symmetric formulations are studied in some detail, and a number of convex and nonconvex applications are given.This work was partially funded by National Research Council of Canada, Grant No. A4493. Thanks are due to Mr. B. Toulany for many conversations and to Dr. L. MacLean who suggested the chance-constrained model.     

Lower bounds on the minimum average distance of binary codes

Let β(n,M) denote the minimum average Hamming distance of a binary code of length n and cardinality M. In this paper we consider lower bounds on β(n,M). All the known lower bounds on β(n,M) are useful when M is at least of size about 2ⁿ⁻¹/n. We derive new lower bounds which give good estimations when size of M is about n. These bounds are obtained using a linear programming approach. In particular, it is proved that lim_n→∞β(n,2n)=5/2. We also give a new recursive inequality for β(n,M).     

On Linear Programming Duality and Necessary and Sufficient Conditions in Minimax Theory

In this paper we discuss necessary and sufficient conditions for different minimax results to hold using only linear programming duality and the finite intersection property for compact sets. It turns out that these necessary and sufficient conditions have a clear interpretation within zero-sum game theory. We apply these results to derive necessary and sufficient conditions for strong duality for a general class of optimization problems. The authors like to thank the comments of the anonymous referees for their remarks, which greatly improved the presentation of this paper.     

Domination analysis for minimum multiprocessor scheduling

Let P be a combinatorial optimization problem, and let A be an approximation algorithm for P. The domination ratio domr(A,s) is the maximal real q such that the solution x(I) obtained by A for any instance I of P of size s is not worse than at least the fraction q of the feasible solutions of I. We say that P admits an asymptotic domination ratio one (ADRO) algorithm if there is a polynomial time approximation algorithm A for P such that . Alon, Gutin and Krivelevich [Algorithms with large domination ratio, J. Algorithms 50 (2004) 118-131] proved that the partition problem admits an ADRO algorithm. We extend their result to the minimum multiprocessor scheduling problem.     

Proving Strong Duality for Geometric Optimization Using a Conic Formulation

Geometric optimization¹ is an important class of problems that has many applications, especially in engineering design. In this article, we provide new simplified proofs for the well-known associated duality theory, using conic optimization. After introducing suitable convex cones and studying their properties, we model geometric optimization problems with a conic formulation, which allows us to apply the powerful duality theory of conic optimization and derive the duality results valid for geometric optimization.     

The minimum Manhattan network problem: Approximations and exact solutions

Given a set of points in the plane and a constant t1, a Euclidean t-spanner is a network in which, for any pair of points, the ratio of the network distance and the Euclidean distance of the two points is at most t. Such networks have applications in transportation or communication network design and have been studied extensively.

In this paper we study 1-spanners under the Manhattan (or L₁-) metric. Such networks are called Manhattan networks. A Manhattan network for a set of points is a set of axis-parallel line segments whose union contains an x- and y-monotone path for each pair of points. It is not known whether it is NP-hard to compute minimum Manhattan networks (MMN), i.e., Manhattan networks of minimum total length. In this paper we present an approximation algorithm for this problem. Given a set P of n points, our algorithm computes in O(nlogn) time and linear space a Manhattan network for P whose length is at most 3 times the length of an MMN of P.

We also establish a mixed-integer programming formulation for the MMN problem. With its help we extensively investigate the performance of our factor-3 approximation algorithm on random point sets.     

Global optimization of a class of nonconvex quadratically constrained quadratic programming problems

In this paper we study a class of nonconvex quadratically constrained quadratic programming problems generalized from relaxations of quadratic assignment problems. We show that each problem is polynomially solved. Strong duality holds if a redundant constraint is introduced. As an application, a new lower bound is proposed for the quadratic assignment problem.     

Multilinear programming: Duality theories

It has been pointed out that the text and proof of a proposition in Ref. 1 are worded ambiguously. The version below is intended to be clearer.     

Multilinear programming: Duality theories

A type of nonlinear programming problem, called multilinear, whose objective function and constraints involve the variables through sums of products is treated. It is a rather straightforward generalization of the linear programming problem. This, and the fact that such problems have recently been encountered in several fields of application, suggested their study, with particular emphasis on the analogies between them and linear problems. This paper develops one such analogy, namely a duality concept which includes its linear counterpart as a special case and also retains essentially all of the desirable characteristics of linear duality theory. It is, however, found that a primal then has several duals. The duals are arrived at by way of a game which is closely associated with a multilinear programming problem, but which differs in some respects from those generally treated in game theory. Its generalizations may in fact be of interest in their own right.Professor J. Stoer and an anonymous reviewer made helpful comments on an earlier version of this paper. Those comments are greatly appreciated.     

On the minimum corridor connection problem and other generalized geometric problems

In this paper we discuss the complexity and approximability of the minimum corridor connection problem where, given a rectilinear decomposition of a rectilinear polygon into “rooms”, one has to find the minimum length tree along the edges of the decomposition such that every room is incident to a vertex of the tree. We show that the problem is strongly NP-hard and give a subexponential time exact algorithm. For the special case when the room connectivity graph is k-outerplanar the algorithm running time becomes cubic. We develop a polynomial time approximation scheme for the case when all rooms are fat and have nearly the same size. When rooms are fat but are of varying size we give a polynomial time constant factor approximation algorithm.     

Helical torsion springs for minimum weight by geometric programming

Geometric programming is applied to solve a design optimization problem for minimum weight of torsional coil springs. An explicit solution of the optimization problem is obtained and applied to a numerical example.The author is indebted to Professor E. J. Haug, University of Iowa, Iowa City, Iowa, for the technical editing of this paper.