We present a class of linear programming approximations for mixed-integer polynomial optimization problems that take advantage of structured sparsity of the constraint matrix. In particular, we show that if the intersection graph of the constraints has tree-width bounded by a constant, then for any desired tolerance there is a linear programming formulation of polynomial size. Via an additional reduction, we obtain a polynomial-time approximation scheme for the "AC-OPF" problem on graphs with bounded tree-width. These constructions partly rely on a general construction for pure binary optimization problems where individual constraints are available through a membership oracle; if the intersection graph for the constraints has bounded tree-width our construction is of linear size and exact. This improves on a number of results in the literature, both from the perspective of formulation size and generality.

We study a Lagrangian decomposition algorithm recently proposed by Dan Bienstock and Mark Zuckerberg for solving the LP relaxation of a class of open pit mine project scheduling problems. In this study we show that the Bienstock-Zuckerberg (BZ) algorithm can be used to solve LP relaxations corresponding to a much broader class of scheduling problems, including the well-known Resource Constrained Project Scheduling Problem (RCPSP), and multi-modal variants of the RCPSP that consider batch processing of jobs. We present a new, intuitive proof of correctness for the BZ algorithm that works by casting the BZ algorithm as a column generation algorithm. This analysis allows us to draw parallels with the well-known Dantzig-Wolfe decomposition (DW) algorithm. We discuss practical computational techniques for speeding up the performance of the BZ and DW algorithms on project scheduling problems. Finally, we present computational experiments independently testing the effectiveness of the BZ and DW algorithms on different sets of publicly available test instances. Our computational experiments confirm that the BZ algorithm significantly outperforms the DW algorithm for the problems considered. Our computational experiments also show that the proposed speed-up techniques can have a significant impact on the solve time. We provide some insights on what might be explaining this significant difference in performance.

Cutting planes are derived from specific problem structures, such as a single linear constraint from an integer program. This paper introduces cuts that involve minimal structural assumptions, enabling the generation of strong polyhedral relaxations for a broad class of problems. We consider valid inequalities for the set S ∩ P , where S is a closed set, and P is a polyhedron. Given an oracle that provides the distance from a point to S we construct a pure cutting plane algorithm; if the initial relaxation is a polytope, the algorithm is shown to converge. Cuts are generated from convex forbidden zones, or S-free sets derived from the oracle. We also consider the special case of polynomial optimization. Polynomial optimization may be represented using a symmetric matrix of variables, and in this lifted representation we can let S be the set of matrices that are real, symmetric outer products. Accordingly we develop a theory of outer-product-free sets. All maximal outer-product-free sets of full dimension are shown to be convex cones and we identify two families of such sets. These families can be used to generate intersection cuts that can separate any infeasible extreme point of a linear programming relaxation in polynomial time. Moreover, in the special case of polynomial optimization we derive strengthened oracle-based intersection cuts that can also ensure separation in polynomial time.arXiv:1610.04604v5 [math.OC]

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