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Maintenance activities help prevent costly generator breakdowns but because generators under maintenance are typically unavailable, the impact of maintenance schedules is significant and their cost must be accounted for when planning maintenance. In this paper, we address the generator maintenance scheduling problem in hydropower systems. We propose a mixed-integer programing model that considers the time windows of the maintenance activities, as well as the nonlinearities and disjunctions of the hydroelectric production functions. Because the resulting model is hard to solve, we also propose an extended formulation, a set reduction approach that uses logical conditions for excluding unnecessary set elements from the model, and valid inequalities. We performed computational experiments using a variety of instances adapted from a real hydropower system in Canada, and the extended formulation with set reduction achieved the best results in terms of computational time and optimality gap. [ABSTRACT FROM AUTHOR]

This paper is concerned with the single-row facility layout problem (SRFLP). A globally optimal solution to the SRFLP is a linear placement of rectangular facilities with varying lengths that achieves the minimum total cost associated with the (known or projected) interactions between them. We demonstrate that the combination of a semidefinite programming relaxation with cutting planes is able to compute globally optimal layouts for large SRFLPs with up to 30 facilities. In particular, we report the globally optimal solutions for two sets of SRFLPs previously studied in the literature, some of which have remained unsolved since 1988. [ABSTRACT FROM AUTHOR]

The article focuses on several studies related to the concepts of satisfiability, Boolean modeling and computation. One study described disjunctive normal form (DNF) tautologies with terms conflicting in one or two variables pairwise and possessing a tree-like structure. Another study proposed a semidefinite programming relaxation for the satisfiability problem. A third study discussed theoretical and experimental results on the use of satisfiability problem (SAT) algorithms for colouring special classes of graphs.

We revisit a regularization technique of Mészáros for handling free variables within interior-point methods for conic linear optimization. We propose a simple computational strategy, supported by a global convergence analysis, for handling the regularization. Using test problems from benchmark suites and recent applications, we demonstrate that the modern code SDPT3 modified to incorporate the proposed regularization is able to achieve the same or significantly better accuracy over standard options of splitting variables, using a quadratic cone, and solving indefinite systems. [ABSTRACT FROM AUTHOR]

We present a new framework for efficiently finding competitive solutions for the facility-layout problem. This framework is based on the combination of two new mathematical-programming models. The first model is a relaxation of the layout problem and is intended to find good starting points for the iterative algorithm used to solve the second model. The second model is an exact formulation of the facility-layout problem as a nonconvex mathematical program with equilibrium constraints (MPEC). Aspect ratio constraints, which are frequently used in facility-layout methods to restrict the occurrence of overly long and narrow departments in the computed layouts, are easily incorporated into this new framework. Finally, we present computational results showing that the complete framework can be solved efficiently using widely available optimization software, and the resulting layouts improve on those obtained using previous approaches in the literature. Moreover, the framework can be used to find different competitive layouts with relatively little computational effort, which is advantageous for a user who wishes to consider several competitive layouts rather than simply using a mathematically optimal layout. [ABSTRACT FROM AUTHOR]

The satisfiability (SAT) problem is a central problem in mathematical logic, computing theory, and artificial intelligence. An instance of SAT is specified by a set of boolean variables and a propositional formula in conjunctive normal form. Given such an instance, the SAT problem asks whether there is a truth assignment to the variables such that the formula is satisfied. It is well known that SAT is in general NP-complete, although several important special cases can be solved in polynomial time. Semidefinite programming (SDP) refers to the class of optimization problems where a linear function of a matrix variableXis maximized (or minimized) subject to linear constraints on the elements ofXand the additional constraint thatXbe positive semidefinite. We are interested in the application of SDP to satisfiability problems, and in particular in how SDP can be used to detect unsatisfiability. In this paper we introduce a new SDP relaxation for the satisfiability problem. This SDP relaxation arises from the recently introduced paradigm of “higher liftings” for constructing semidefinite programming relaxations of discrete optimization problems. To derive the SDP relaxation, we first formulate SAT as an optimization problem involving matrices. Relaxing this formulation yields an SDP which significantly improves on the previous relaxations in the literature. The important characteristics of the SDP relaxation are its ability to prove that a given SAT formula is unsatisfiable independently of the lengths of the clauses in the formula, its potential to yield truth assignments satisfying the SAT instance if a feasible matrix of sufficiently low rank is computed, and the fact that it is more amenable to practical computation than previous SDPs arising from higher liftings. We present theoretical and computational results that support these claims. [ABSTRACT FROM AUTHOR]

This paper is concerned with the analysis and comparison of semidefinite programming (SDP) relaxations for the satisfiability (SAT) problem. Our presentation is focussed on the special case of 3-SAT, but the ideas presented can in principle be extended to any instance of SAT specified by a set of boolean variables and a propositional formula in conjunctive normal form. We propose a new SDP relaxation for 3-SAT and prove some of its theoretical properties. We also show that, together with two SDP relaxations previously proposed in the literature, the new relaxation completes a trio of linearly sized relaxations with increasing rank-based guarantees for proving satisfiability. A comparison of the relative practical performances of the SDP relaxations shows that, among these three relaxations, the new relaxation provides the best tradeoff between theoretical strength and practical performance within an enumerative algorithm. [ABSTRACT FROM AUTHOR]

In this paper we study two strengthened semidefinite programming relaxations for the Max-Cut problem. Our results hold for every instance of Max-Cut; in particular, we make no assumptions about the edge weights. We prove that the first relaxation provides a strengthening of the Goemans–Williamson relaxation. The second relaxation is a further tightening of the first one and we prove that its feasible set corresponds to a convex set that is larger than the cut polytope but nonetheless is strictly contained in the intersection of the elliptope and the metric polytope. Both relaxations are obtained using Lagrangian relaxation. Hence, our results also exemplify the strength and flexibility of Lagrangian relaxation for obtaining a variety of SDP relaxations with different properties.We also address some practical issues in the solution of these SDP relaxations. Because Slater''s constraint qualification fails for both of them, we project their feasible sets onto a lower dimensional space in a way that does not affect the sparsity of these relaxations but guarantees Slater''s condition. Some preliminary numerical results are included. [Copyright &y& Elsevier]