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Register a new userCompact linear programs for 2SAT
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Added by
Hans Raj Tiwary
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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For each integer $n$ we present an explicit formulation of a compact linear program, with $O(n^3)$ variables and constraints, which determines the satisfiability of any 2SAT formula with $n$ boolean variables by a single linear optimization. This contrasts with the fact that the natural polytope for this problem, formed from the convex hull of all satisfiable formulas and their satisfying assignments, has superpolynomial extension complexity. Our formulation is based on multicommodity flows. We also discuss connections of these results to the stable matching problem.
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Inspired by the decomposition in the hybrid quantum-classical optimization algorithm we introduced in arXiv:1902.04215, we propose here a new (fully classical) approach to solving certain non-convex integer programs using Graver bases. This method is well suited when (a) the constraint matrix $A$ has a special structure so that its Graver basis can be computed systematically, (b) several feasible solutions can also be constructed easily and (c) the objective function can be viewed as many convex functions quilted together. Classes of problems that satisfy these conditions include Cardinality Boolean Quadratic Problems (CBQP), Quadratic Semi-Assignment Problems (QSAP) and Quadratic Assignment Problems (QAP). Our Graver Augmented Multi-seed Algorithm (GAMA) utilizes augmentation along Graver basis elements (the improvement direction is obtained by comparing objective function values) from these multiple initial feasible solutions. We compare our approach with a best-in-class commercially available solver (Gurobi). Sensitivity analysis indicates that the rate at which GAMA slows down as the problem size increases is much lower than that of Gurobi. We find that for several instances of practical relevance, GAMA not only vastly outperforms in terms of time to find the optimal solution (by two or three orders of magnitude), but also finds optimal solutions within minutes when the commercial solver is not able to do so in 4 or 10 hours (depending on the problem class) in several cases.
We analyze a distributed approach for automatically reconfiguring distribution systems into an operational radial network after a fault occurs by creating an ordering in which switches automatically close upon detection of a downstream fault. The switches reconnection ordering significantly impacts the expected time to reconnect under normal disruptions and thus affects reliability metrics such as SAIDI and CAIDI, which are the basis for regulator-imposed financial incentives for performance. We model the problem of finding a switch reconnection ordering that minimizes SAIDI and the expected reconnection time as Minimum Reconnection Time (MRT), which we show is a special case of the well-known minimum linear ordering problem from the submodular optimization literature, and in particular the Min Sum Set Cover problem (MSSC). We prove that MRT is also NP-hard. We generalize the kernel-based rounding approaches of Bansal et al. for Min Sum Vertex Cover to give tight approximation guarantees for MSSC on c-uniform hypergraphs for all c. For all instances of MSSC, our methods have a strictly better approximation ratio guarantee than the best possible methods for general MSSC. Finally, we consider optimizing multiple metrics simultaneously using local search methods that also reconfigure the systems base tree to ensure fairness in service disruptions and reconnection times and reduce energy loss. We computationally validate our approach on the NREL SMART-DS Greensboro synthetic urban-suburban network. We evaluate the performance of our reconfiguration methods and show significant reductions compared to single-metric-based optimizations.
We derive equivalent linear and dynamic programs for infinite horizon risk-sensitive control for minimization of the asymptotic growth rate of the cumulative cost.
We settle the computational complexity of fundamental questions related to multicriteria integer linear programs, when the dimensions of the strategy space and of the outcome space are considered fixed constants. In particular we construct: 1. polynomial-time algorithms to exactly determine the number of Pareto optima and Pareto strategies; 2. a polynomial-space polynomial-delay prescribed-order enumeration algorithm for arbitrary projections of the Pareto set; 3. an algorithm to minimize the distance of a Pareto optimum from a prescribed comparison point with respect to arbitrary polyhedral norms; 4. a fully polynomial-time approximation scheme for the problem of minimizing the distance of a Pareto optimum from a prescribed comparison point with respect to the Euclidean norm.
Motivated by a growing list of nontraditional statistical estimation problems of the piecewise kind, this paper provides a survey of known results supplemented with new results for the class of piecewise linear-quadratic programs. These are linearly constrained optimization problems with piecewise linear-quadratic (PLQ) objective functions. Starting from a study of the representation of such a function in terms of a family of elementary functions consisting of squared affine functions, squared plus-composite-affine functions, and affine functions themselves, we summarize some local properties of a PLQ function in terms of their first and second-order directional derivatives. We extend some well-known necessary and sufficient second-order conditions for local optimality of a quadratic program to a PLQ program and provide a dozen such equivalent conditions for strong, strict, and isolated local optimality, showing in particular that a PLQ program has the same characterizations for local minimality as a standard quadratic program. As a consequence of one such condition, we show that the number of strong, strict, or isolated local minima of a PLQ program is finite; this result supplements a recent result about the finite number of directional stationary objective values. Interestingly, these finiteness results can be uncovered by invoking a very powerful property of subanalytic functions; our proof is fairly elementary, however. We discuss applications of PLQ programs in some modern statistical estimation problems. These problems lead to a special class of unconstrained composite programs involving the non-differentiable $ell_1$-function, for which we show that the task of verifying the second-order stationary condition can be converted to the problem of checking the copositivity of certain Schur complement on the nonnegative orthant.
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