Second-Order Cone Relaxations for Binary Quadratic Polynomial Programs

SIAM Journal on Optimization (Impact Factor: 1.83). 01/2011; 21(1):391-414. DOI: 10.1137/100802190
Source: DBLP


techniques. In this paper, we propose a general framework to construct conic relaxations for binary quadratic polynomial programs based on polynomial programming. Using our frame- work, we re-derive previous relaxation schemes and provide new ones. In particular, we present three relax- ations for binary quadratic polynomial programs. The rst two relaxations, based on second-order cone and semidenite programming, represent a signicant improvement over previous practical relaxations for several classes of non-convex binary quadratic polynomial problems. From a practical point of view, due to the computational cost, semidenite-based relaxations for binary quadratic polynomial problems can be used only to solve small to mid-size instances. To improve the computational eciency for solving such problems, we propose a third relaxation based purely on second-order cone programming. Computational tests on dif- ferent classes of non-convex binary quadratic polynomial problems, including quadratic knapsack problems, show that the second-order cone-based relaxation outperforms the semidenite-based relaxations that are proposed in the literature in terms of computational eciency and is comparable in terms of bounds.

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Available from: Juan C Vera, Jan 21, 2014
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    • "However, the SDP constraints are computationally expensive and thus even using low-orders of the hierarchy to approximate large-scale polynomial optimization problems becomes computationally intractable in practice [22]. To improve the computational performance of the SDP based hierarchies to approximate the solution of POPs, prior work has focused on exploiting the problem structure through sparsity [19] and symmetry [9] [10], improving the relaxation through generating valid inequalities [14], and more recently through devising more computationally efficient hierarchies such as linear programming (LP) and second-order cone programming (SOCP) hierarchies [1] [13] [28]. Here, we consider alternative ways to use SOCP restrictions of the SOS condition introduced in Ahmadi and Majumdar [1]. "
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    ABSTRACT: In theory, hierarchies of semidefinite programming (SDP) relaxations based on sum of squares (SOS) polynomials have been shown to provide arbitrarily close approximations to general polynomial optimization problems (POP). However, due to the computational challenge of solving SDPs, it becomes difficult to use SDP hierarchies for large-scale problems. To address this, hierarchies of second-order cone programming (SOCP) relaxations resulting from a restriction of the SOS polynomial condition have been recently proposed to approximate POPs. Here, we consider alternative ways to use these SOCP restriction of the SOS condition. In particular, we show that SOCP hierarchies can be effectively used to strengthen hierarchies of linear programming (LP) relaxations for POPs. Specifically, we show that this solution approach is substantially more effective in finding solutions of certain POPs for which the more common hierarchies of SDP relaxations are known to perform poorly. Also, we use hierarchies of LP relaxations for POPs that allows us to show that the SOCP approach can be used to obtain hierarchies of SOCPs that converge to the optimal value of the POP when its feasible set is compact. Additionally, we show that the SOCP approach can be used to address the solution of the fundamental alternating current optimal power flow (ACOPF) problem. In particular, we show that the first-order SOCP hierarchy obtained by weakening the more common hierarchy of SDP relaxations for this problem is equivalent to solving the conic dual of the SOCP approximations recently proposed to address the ACOPF problem. Through out the article, we illustrate our findings with relevant experimental results. In the case of the ACOPF problem, we use well-known instances of the problem that appear in the related literature.
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    • "Problem (7) is one such example that leads to a somewhat concise semi-definite program. We refer the interested reader to Lasserre (2002) and Ghaddar et al. (2011) for a hierarchy of polynomial size semi-definite programming relaxation of mixed-integer quadratic programs for which the integrality gap is known to converge to 1. Based on Proposition 2, it is therefore theoretically possible to find a semi-definite programming model of polynomial size that will generate a solution within a constant factor of the optimal one. Unfortunately, this might often be of little practical relevance. "
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    ABSTRACT: Robust optimization is a methodology that has gained a lot of attention in the recent years. This is mainly due to the simplicity of the modeling process and ease of resolution even for large scale models. Unfortunately, the second property is usually lost when the cost function that needs to be “robustified” is not concave (or linear) with respect to the perturbing parameters. In this paper, we study robust optimization of sums of piecewise linear functions over polyhedral uncertainty set. Given that these problems are known to be intractable, we propose a new scheme for constructing conservative approximations based on the relaxation of an embedded mixed-integer linear program and relate this scheme to methods that are based on exploiting affine decision rules. Our new scheme gives rise to two tractable models that respectively take the shape of a linear program and a semi-definite program, with the latter having the potential to provide solutions of better quality than the former at the price of heavier computations. We present conditions under which our approximation models are exact. In particular, we are able to propose the first exact reformulations for a robust (and distributionally robust) multi-item newsvendor problem with budgeted uncertainty set and a reformulation for robust multi-period inventory problems that is exact whether the uncertainty region reduces to a L1-norm ball or to a box. An extensive set of empirical results will illustrate the quality of the approximate solutions that are obtained using these two models on randomly generated instances of the latter problem.
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    • "Linear refers to generating linear inequalities that are added to the master problem by using a non-negative multiplier. SOCP refers to generating linear inequalities that are added to the master problem by using a polynomial multiplier that is in P 1 (B) [10]. Quadratic refers to generating quadratic inequalities similar to the previous examples described. "
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    ABSTRACT: Semidefinite programming has been used successfully to build hierarchies of convex relaxations to approximate polynomial programs. This approach rapidly becomes computationally expensive and is often tractable only for problems of small sizes. We propose an iterative scheme that improves the semidefinite relaxations without incurring exponential growth in their size. The key ingredient is a dynamic scheme for generating valid polynomial inequalities for general polynomial programs. These valid inequalities are then used to construct better approximations of the original problem. As a result, the proposed scheme is in principle scalable to large general combinatorial optimization problems. For binary polynomial programs, we prove that the proposed scheme converges to the global optimal solution for interesting cases of the initial approximation of the problem. We also present examples illustrating the computational behaviour of the scheme and compare it to other methods in the literature.
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