An interior multiobjective primal-dual linear programming algorithm using approximated gradients and sequential generation of anchor points

An algorithm for addressing multiple objective linear programming (MOLP) problems is presented. The algorithm modifies the path-following primal-dual algorithm to MOLP problems by using the single objective algorithm to generate interior search directions and later combine them to derive a single direction along which to step to the next iterate. Combining the different interior search directions is done by interacting with a Decision Maker (DM) to obtain locally-relevant preference information for the value vectors along these directions. This preference information is then used to derive an approximation to the gradient of an implicity-known utility function, and using a projection of this gradient provides a direction gradient of an implicitly-known utility function, and using a projection of this gradient provides a direction vector along which we step to the next iterate. At each iteration the algorithm also generates boundary points that aid in deriving the combined search direction. We refer to these boundary points, generated sequentially during the process, as anchor points that serve as candidate solutions at which to terminate the iterative process.     

A 2 + ɛ approximation algorithm for the k-MST problem

For any ɛ > 0 we give a (2 + ɛ)-approximation algorithm for the problem of finding a minimum tree spanning any k vertices in a graph (k-MST), improving a 3-approximation algorithm by Garg [10]. As in [10] the algorithm extends to a (2 + ɛ)-approximation algorithm for the minimum tour that visits any k vertices, provided the edge costs satisfy the triangle inequality. Research supported by NSF CAREER award NSF CCR-9502747, NSF grants CCR-0205594 and CCR-0098180, an Alfred Sloan Fellowship, and a Packard Fellowship. Research supported by an NSERC Discovery grant.     

The minimum cost flow problem: A unifying approach to dual algorithms and a new tree-search algorithm

This paper is concerned with the minimum cost flow problem. It is shown that the class of dual algorithms which solve this problem consists of different variants of a common general algorithm. We develop a new variant which is, in fact, a new form of the ‘primal-dual algorithm’ and which has several interesting properties. It uses, explicitly only dual variables. The slope of the change in the (dual) objective is monotone. The bound on the maximum number of iterations to solve a problem with integral bounds on the flow is better than bounds for other algorithms. This paper is part of the author's doctoral dissertation submitted at Yale University.     

A unified framework for primal-dual methods in minimum cost network flow problems

We introduce a broad class of algorithms for finding a minimum cost flow in a capacitated network. The algorithms are of the primal-dual type. They maintain primal feasibility with respect to capacity constraints, while trying to satisfy the conservation of flow equation at each node by means of a wide variety of procedures based on flow augmentation, price adjustment, and ascent of a dual functional. The manner in which these procedures are combined is flexible thereby allowing the construction of algorithms that can be tailored to the problem at hand for maximum effectiveness. Particular attention is given to methods that incorporate features from classical relaxation procedures. Experimental codes based on these methods outperform by a substantial margin the fastest available primal-dual and primal simplex codes on standard benchmark problems.This work was supported by the National Science Foundation under Contract NSF/ECS 8217668.     

参数最短路的原始-对偶算法

本文利用原始－对偶方法,对于含参数λ的网络（Ｖ,Ｅ,ｆ_１-λｆ_２）,给出了某一点至其它各点的参数最短路的求解算法,其时间复杂度为 Ｏ（ｎｍ＋ｎ~２ｌｏｇｎ）．     

On the stability of general convex programs under slater’s condition and primal solution boundedness

This paper was motivated by an article by Best and Chakravarti, who presented some stability results for convex quadratic programs under linear perturbation of the data. We show that the regularity conditions assumed are much too restrictive and demonstrate that stronger stability results follow under weaker assumptions (primal solution boundedness and the Slater condition) and from known results, not only for convex quadratic problems but for general convex programs with general perturbations. In so doing, we give a simple and reasonably complete characterization of the stability of an important class of well-behaved convex programs, collecting results that heretofore have apparently not been presented in a unified manner. The results, virtually all from Hogan and Robinson, involve mainly stability of the feasible region and solution existence under small perturbations, and continuity and differentiability of the optimal value function. We note that Auslender and Coutat have recently provided similar extensions for saddle points of generalized linear-quadratic programs introduced by Rockafellar and Wets, utilizing the same assumptions that we use in this paper