A practical relative error criterion for augmented Lagrangians

This paper develops a new error criterion for the approximate minimization of augmented Lagrangian subproblems. This criterion is practical since it is readily testable given only a gradient (or subgradient) of the augmented Lagrangian. It is also “relative” in the sense of relative error criteria for proximal point algorithms: in particular, it uses a single relative tolerance parameter, rather than a summable parameter sequence. Our analysis first describes an abstract version of the criterion within Rockafellar’s general parametric convex duality framework, and proves a global convergence result for the resulting algorithm. Specializing this algorithm to a standard formulation of convex programming produces a version of the classical augmented Lagrangian method with a novel inexact solution condition for the subproblems. Finally, we present computational results drawn from the CUTE test set—including many nonconvex problems—indicating that the approach works well in practice.     

The detection and estimation of the change point in a disccrete-time stochastic system

A piecewise-constant process containing a single jump is observed under noise in the context of discrete time. The conditional density and maximum a posteriori (MAP) estimator of the jump time as well as the Bayes detector of the jump itself are determined using the powerful measure transformation approach. The Bayes detector provides a convenient sequential detection rule for practical on-line implementation. An asymptotic result for the distribution of the MAP estimator's estimation error and the corresponding convergence rate are derived. This result provides a reference measure of optimal performance for jump-time estimators in discrete-time stochastic systems that does not depend on the jump time's prior distribution     

Matroid optimisation problems with nested non-linear monomials in the objective function

We derive a new approximate version of the alternating direction method of multipliers (ADMM) which uses a relative error criterion. The new version is somewhat restrictive and allows only one of the two subproblems to be minimized approximately, but nevertheless covers commonly encountered special cases. The derivation exploits the long-established relationship between the ADMM and both the proximal point algorithm (PPA) and Douglas–Rachford (DR) splitting for maximal monotone operators, along with a relative-error of the PPA due to Solodov and Svaiter. In the course of analysis, we also derive a version of DR splitting in which one operator may be evaluated approximately using a relative error criterion. We computationally evaluate our method on several classes of test problems and find that it significantly outperforms several alternatives on one problem class.     

On acyclicity of games with cycles

We study restricted improvement cycles (ri-cycles) in finite positional n-person games with perfect information modeled by directed graphs (di-graphs) that may contain directed cycles (di-cycles). We assume that all these di-cycles form one outcome c, for example, a draw. We obtain criteria of restricted improvement acyclicity (ri-acyclicity) in two cases: for n=2 and for acyclic di-graphs. We provide several examples that outline the limits of these criteria and show that, essentially, there are no other ri-acyclic cases.We also discuss connections between ri-acyclicity and some open problems related to Nash-solvability.     

The Maximum Box Problem and its Application to Data Analysis

Given two finite sets of points X ⁺ and X ^– in ⁿ , the maximum box problem consists of finding an interval (box) B = {x : l x u} such that B X ^– = , and the cardinality of B X ⁺ is maximized. A simple generalization can be obtained by instead maximizing a weighted sum of the elements of B X ⁺. While polynomial for any fixed n, the maximum box problem is -hard in general. We construct an efficient branch-and-bound algorithm for this problem and apply it to a standard problem in data analysis. We test this method on nine data sets, seven of which are drawn from the UCI standard machine learning repository.     

On the probability of union in the n-space

In this paper, our sets are orthants in $R^n$ and $N$, the number of them, is large ($N>n$). We introduce the modified inclusion–exclusion formula in order to efficiently calculate the probability of a union of such events. The new formula works in the bivariate case, and can also be used in $R^n,n\ge 3$ with a condition on the projected sets onto lower dimensional spaces. Numerical examples are presented.     

Roof duality,complementation and persistency in quadratic 0–1 optimization

The paper is concerned with the ‘primal’ problem of maximizing a given quadratic pseudo-boolean function. Four equivalent problems are discussed—the primal, the ‘complementation’, the ‘discrete Rhys LP’ and the ‘weighted stability problem of a SAM graph’. Each of them has a relaxation—the ‘roof dual’, the ‘quadratic complementation,’ the ‘continuous Rhys LP’ and the ‘fractional weighted stability problem of a SAM graph’. The main result is that the four gaps associated with the four relaxations are equal. Furthermore, a solution to any of these problems leads at once to solutions of the other three equivalent ones. The four relaxations can be solved in polynomial time by transforming them to a bipartite maximum flow problem. The optimal solutions of the ‘roof-dual’ define ‘best’ linear majorantsp(x) off, having the following persistency property: if theith coefficient inp is positive (negative) thenx _i=1 (0) in every optimum of the primal problem. Several characterizations are given for the case where these persistency results cannot be used to fix any variable of the primal. On the other hand, a class of gap-free functions (properly including the supermodular ones) is exhibited.     

组合最优化中的布尔方法(续二)

§13 天篷取优 为了计算最好的天篷,我们研究在SAM-图S_f上的最大W-对集问题(WM)。首先,对于S_f上的边引进变量:     

The compromise value for NTU-games

The compromise value is introduced as a single-valued solution concept for NTU-games. It is shown that the compromise value coincides with the -value for TU-games and with the Kalai-Smorodinsky solution for bargaining problems. In addition the axiomatic characterizations of both the two-person Kalai-Smorodinsky solution and the -value can be extended to the compromise value for large classes of NTU-games.We also present an alternative NTU-extension of the TU -value (called the NTU -value) which coincides with the Nash solution for two-person bargaining problems. The definition of the NTU -value is analogous to that of the Shapley NTU-value.     

Concavity and efficient points of discrete distributions in probabilistic programming

We consider stochastic programming problems with probabilistic constraints involving integer-valued random variables. The concept of a p-efficient point of a probability distribution is used to derive various equivalent problem formulations. Next we introduce the concept of r-concave discrete probability distributions and analyse its relevance for problems under consideration. These notions are used to derive lower and upper bounds for the optimal value of probabilistically constrained stochastic programming problems with discrete random variables. The results are illustrated with numerical examples. Received: October 1998 / Accepted: June 2000?Published online October 18, 2000