On a connection between facility location and perfect graphs

We characterize the graphs for which a linear relaxation of a facility location problem defines a polytope with all integral extreme points. We use a transformation to a stable set problem in perfect graphs. Based on this transformation, these graphs can be recognized in polynomial time.     

On a geometric property of perfect graphs

LetG be a graph,VP(G) its vertex packing polytope and letA(G) be obtained by reflectingVP(G) in all Cartersian coordinates. Denoting byA*(G) the set obtained similarly from the fractional vertex packing polytope, we prove that the segment connecting any two non-antipodal vertices ofA(G) is contained in the surface ofA(G) and thatG is perfect if and only ifA*(G) has a similar property.     

A new dual problem for nondifferentiable convex programming

This paper gives a new dual problem for nondifferentiable convex programming and proves the properties of weak duality and strong duality and offers a necessary and sufficient condition of strong duality.     

A neural network for solving a convex quadratic bilevel programming problem

A neural network is proposed for solving a convex quadratic bilevel programming problem. Based on Lyapunov and LaSalle theories, we prove strictly an important theoretical result that, for an arbitrary initial point, the trajectory of the proposed network does converge to the equilibrium, which corresponds to the optimal solution of a convex quadratic bilevel programming problem. Numerical simulation results show that the proposed neural network is feasible and efficient for a convex quadratic bilevel programming problem.     

A central cutting plane algorithm for the convex programming problem

An algorithm is developed for solving the convex programming problem which iteratively proceeds to the optimum by constructing a cutting plane through the center of a polyhedral approximation to the optimum. This generates a sequence of primal feasible points whose limit points satisfy the Kuhn—Tucker conditions of the problem. Additionally, we present a simple, effective rule for dropping prior cuts, an easily calculated bound on the objective function, and a rate of convergence.     

Feasible partition problem in reverse convex and convex mixed-integer programming

In this paper we consider the consistent partition problem in reverse convex and convex mixed-integer programming. In particular we will show that for the considered classes of convex functions, both integer and relaxed systems can be partitioned into two disjoint subsystems, each of which is consistent and defines an unbounded region. The polynomial time algorithm to generate the partition will be proposed and the algorithm for a maximal partition will also be provided.     

A note on perfect graphs

Let the lines of a complete graph be 3-colored so that no triangle gets 3 different colors. If two of these colors form perfect graphs then so does the third.     

Duality-based regularization in a linear convex mathematical programming problem

For a linear convex mathematical programming (MP) problem with equality and inequality constraints in a Hilbert space, a dual-type algorithm is constructed that is stable with respect to input data errors. In the algorithm, the dual of the original optimization problem is solved directly on the basis of Tikhonov regularization. It is shown that the necessary optimality conditions in the original MP problem are derived in a natural manner by using dual regularization in conjunction with the constructive generation of a minimizing sequence. An iterative regularization of the dual algorithm is considered. A stopping rule for the iteration process is presented in the case of a finite fixed error in the input data.     

Error bounds for the aggregated convex programming problem

Applied mathematical programming problems are often approximations of larger, more detailed problems. One criterion to evaluate an approximating program is the magnitude of the difference between the optimal objective values of the original and the approximating program. The approximation we consider is variable aggregation in a convex program. Bounds are derived on the difference between the two optimal objective values. Previous results of Geoffrion and Zipkin are obtained by specializing our results to linear programming. Also, we apply our bounds to a convex transportation problem. Thanks are due to Ron Dembo, Paul Zipkin and the referees for valuable comments. This research was supported by NSF Grant ENG-76-15599.     

A global optimization approach for solving the convex multiplicative programming problem

We consider a convex multiplicative programming problem of the form% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGG7bGaam% OzamaaBaaaleaacaaIXaaabeaakiaacIcacaWG4bGaaiykaiabgwSi% xlaadAgadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaamiEaiaacMcaca% GG6aGaamiEaiabgIGiolaadIfacaGG9baaaa!4A08!\[\{ f_1 (x) \cdot f_2 (x):x \in X\} \]where X is a compact convex set of ⁿ and f ₁, f ₂ are convex functions which have nonnegative values over X.Using two additional variables we transform this problem into a problem with a special structure in which the objective function depends only on two of the (n+2) variables. Following a decomposition concept in global optimization we then reduce this problem to a master problem of minimizing a quasi-concave function over a convex set in ² ₂. This master problem can be solved by an outer approximation method which requires performing a sequence of simplex tableau pivoting operations. The proposed algorithm is finite when the functions f _i, (i=1, 2) are affine-linear and X is a polytope and it is convergent for the general convex case.Partly supported by the Deutsche Forschungsgemeinschaft Project CONMIN.     

A new bound for the quadratic assignment problem based on convex quadratic programming

We describe a new convex quadratic programming bound for the quadratic assignment problem (QAP). The construction of the bound uses a semidefinite programming representation of a basic eigenvalue bound for QAP. The new bound dominates the well-known projected eigenvalue bound, and appears to be competitive with existing bounds in the trade-off between bound quality and computational effort. Received: February 2000 / Accepted: November 2000?Published online January 17, 2001     

A polynomial algorithm for the minimum weighted clique cover problem on claw-free perfect graphs

Let G = (V, E) be a graph with a positive number wt(v) assigned to each v ? v. A weighted clique cover of the vertices of G is a collection of cliques with a non-negative weight y_c assigned to each clique C in the collection such that σ_C:v?CY_C?wt(v) for all v ? V. The problem considered is to minimize σ_Cy_C over all weighted clique covers. A polynomial time algorithm for this problem is presented for graphs that are claw-free and perfect.     

A connection between the moment problem and the subnormal completion problem

In this note we give a connection between the truncated complex moment problem and the subnormal completion problem for the unilateral weighted shifts.     

An exterior point method for the convex programming problem

This paper gives a proof of convergence of an iterative method for maximizing a concave function subject to inequality constraints involving convex functions. The linear programming problem is an important special case. The primary feature is that each iteration is very simple computationally, involving only one of the constraints. Although the paper is theoretical in nature, some numerical results are included.The author wishes to express his gratitude to Ms. A. Dunham, who provided a great deal of assistance in carrying out the computations presented in this paper.     

An approximate method for solving the convex programming problem

We consider an iterative process for maximization of a convex nondifferentiable functional in a real Hilbert space. Two-sided bounds on the optimal functional value are derived. Stability of the approximate solutions is considered. Convergence of the proposed iterative process is proved.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 122–129, 1986     

A greedy algorithm for solving a class of convex programming problems and its connection with polymatroid theory

We present a greedy algorithm for solving a special class of convex programming problems and establish a connection with polymatroid theory which yields a theoretical explanation and verification of the algorithm via some recent results of S. Fujishige.