Resolution over Linear Equations and Multilinear Proofs

We develop and study the complexity of propositional proof systems of varying strength extending resolution by allowing it to operate with disjunctions of linear equations instead of clauses. We demonstrate polynomial-size refutations for hard tautologies like the pigeonhole principle, Tseitin graph tautologies and the clique-coloring tautologies in these proof systems. Using the (monotone) interpolation by a communication game technique we establish an exponential-size lower bound on refutations in a certain, considerably strong, fragment of resolution over linear equations, as well as a general polynomial upper bound on (non-monotone) interpolants in this fragment. We then apply these results to extend and improve previous results on multilinear proofs (over fields of characteristic 0), as studied in [RazTzameret06]. Specifically, we show the following: 1. Proofs operating with depth-3 multilinear formulas polynomially simulate a certain, considerably strong, fragment of resolution over linear equations. 2. Proofs operating with depth-3 multilinear formulas admit polynomial-size refutations of the pigeonhole principle and Tseitin graph tautologies. The former improve over a previous result that established small multilinear proofs only for the \emph{functional} pigeonhole principle. The latter are different than previous proofs, and apply to multilinear proofs of Tseitin mod p graph tautologies over any field of characteristic 0. We conclude by connecting resolution over linear equations with extensions of the cutting planes proof system.Comment: 44 page

Lower Bound Techniques for QBF Proof Systems

How do we prove that a false QBF is inded false? How big a proof is needed? The special case when all quantifiers are existential is the well-studied setting of propositional proof complexity. Expectedly, universal quantifiers change the game significantly. Several proof systems have been designed in the last couple of decades to handle QBFs. Lower bound paradigms from propositional proof complexity cannot always be extended - in most cases feasible interpolation and consequent transfer of circuit lower bounds works, but obtaining lower bounds on size by providing lower bounds on width fails dramatically. A new paradigm with no analogue in the propositional world has emerged in the form of strategy extraction, allowing for transfer of circuit lower bounds, as well as obtaining independent genuine QBF lower bounds based on a semantic cost measure. This talk will provide a broad overview of some of these developments

Playing with Duality: An Overview of Recent Primal-Dual Approaches for Solving Large-Scale Optimization Problems

Optimization methods are at the core of many problems in signal/image processing, computer vision, and machine learning. For a long time, it has been recognized that looking at the dual of an optimization problem may drastically simplify its solution. Deriving efficient strategies which jointly brings into play the primal and the dual problems is however a more recent idea which has generated many important new contributions in the last years. These novel developments are grounded on recent advances in convex analysis, discrete optimization, parallel processing, and non-smooth optimization with emphasis on sparsity issues. In this paper, we aim at presenting the principles of primal-dual approaches, while giving an overview of numerical methods which have been proposed in different contexts. We show the benefits which can be drawn from primal-dual algorithms both for solving large-scale convex optimization problems and discrete ones, and we provide various application examples to illustrate their usefulness

Survey on assembly sequencing: a combinatorial and geometrical perspective

A systematic overview on the subject of assembly sequencing is presented. Sequencing lies at the core of assembly planning, and variants include finding a feasible sequence—respecting the precedence constraints between the assembly operations—, or determining an optimal one according to one or several operational criteria. The different ways of representing the space of feasible assembly sequences are described, as well as the search and optimization algorithms that can be used. Geometry plays a fundamental role in devising the precedence constraints between assembly operations, and this is the subject of the second part of the survey, which treats also motion in contact in the context of the actual performance of assembly operations.Peer ReviewedPostprint (author’s final draft

Data-Driven Estimation in Equilibrium Using Inverse Optimization

Equilibrium modeling is common in a variety of fields such as game theory and transportation science. The inputs for these models, however, are often difficult to estimate, while their outputs, i.e., the equilibria they are meant to describe, are often directly observable. By combining ideas from inverse optimization with the theory of variational inequalities, we develop an efficient, data-driven technique for estimating the parameters of these models from observed equilibria. We use this technique to estimate the utility functions of players in a game from their observed actions and to estimate the congestion function on a road network from traffic count data. A distinguishing feature of our approach is that it supports both parametric and \emph{nonparametric} estimation by leveraging ideas from statistical learning (kernel methods and regularization operators). In computational experiments involving Nash and Wardrop equilibria in a nonparametric setting, we find that a) we effectively estimate the unknown demand or congestion function, respectively, and b) our proposed regularization technique substantially improves the out-of-sample performance of our estimators.Comment: 36 pages, 5 figures Additional theorems for generalization guarantees and statistical analysis adde

Approximation schemes for non-separable non-linear Boolean programming problems under nested knapsack constraints

We consider a fairly general model of “take-or-leave”decision-making. Given a number of items of a particular weight, the decision-maker either takes (accepts) an item or leaves (rejects) it. We design fully polynomial-time approximation schemes (FPTASs) for optimization of a non-separable non-linear function which depends on which items are taken and which are left. The weights of the taken items are subject to nested constraints. There is a noticeable lack of approximation results on integer programming problems with non-separable functions. Most of the known positive results address special forms of quadratic functions, and in order to obtain the corresponding approximation algorithms and schemes considerable technical difficulties have to be overcome. We demonstrate how for the problem under consideration and its modifications FPTASs can be designed by using (i) the geometric rounding techniques, and (ii) methods of K -approximation sets and functions. While the latter approach leads to a faster scheme, the running times of the of both algorithms compare favorably with known analogues for less general problems