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Convex relaxations of non-convex mixed integer quadratically constrained programs: Projected formulations
- Mathematical Programming, Series A
, 2009
"... Abstract A common way to produce a convex relaxation of a Mixed Integer Quadratically Constrained Program (MIQCP) is to lift the problem into a higher dimensional space by introducing variables Yij to represent each of the products xixj of variables appearing in a quadratic form. One advantage of su ..."
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Cited by 27 (3 self)
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Abstract A common way to produce a convex relaxation of a Mixed Integer Quadratically Constrained Program (MIQCP) is to lift the problem into a higher dimensional space by introducing variables Yij to represent each of the products xixj of variables appearing in a quadratic form. One advantage of such extended relaxations is that they can be efficiently strengthened by using the (convex) SDP constraint Y − xx T ≽ 0 and disjunctive programming. On the other hand, their main drawback is their huge size, even for problems of moderate size. In this paper, we study methods to build low-dimensional relaxations of MIQCP that capture the strength of the extended formulations. To do so, we use projection techniques pioneered in the context of the lift-and-project methodology. We show how the extended formulation can be algorithmically projected to the original space by solving linear programs. Furthermore, we extend the technique to project the SDP relaxation by solving SDPs. In the case of an MIQCP with a single quadratic constraint, we propose a subgradient-based heuristic to efficiently solve these SDPs. We also propose a new eigen reformulation for MIQCP, and a cut generation technique to strengthen this reformulation using polarity. We present extensive computational results to illustrate the efficiency of the proposed techniques. Our computational results have two highlights. First, on the GLOBALLib instances, we are able to generate relaxations that are almost
Conic mixed-integer rounding cuts
- University of California-Berkeley
, 2006
"... Abstract. A conic integer program is an integer programming problem with conic constraints. Many important problems in finance, engineering, statistical learning, and probabilistic optimization are modeled using conic constraints. Here we study mixed-integer sets defined by second-order conic constr ..."
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Cited by 21 (4 self)
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Abstract. A conic integer program is an integer programming problem with conic constraints. Many important problems in finance, engineering, statistical learning, and probabilistic optimization are modeled using conic constraints. Here we study mixed-integer sets defined by second-order conic constraints. We introduce general-purpose cuts for conic mixed-integer programming based on polyhedral conic substructures of second-order conic sets. These cuts can be readily incorporated in branch-and-bound algorithms that solve continuous conic programming or linear programming relaxations of conic integer programs at the nodes of the branch-and-bound tree. Central to our approach is a reformulation of the second-order conic constraints with polyhedral second-order conic constraints in a higher dimensional space. In this representation the cuts we develop are linear, even though they are nonlinear in the original space of variables. This feature leads to computationally efficient implementation of nonlinear cuts for conic mixed-integer programming. The reformulation also allows the use of polyhedral methods for conic integer programming. Our computational experiments show that conic mixed-integer rounding cuts are very effective in reducing the integrality gap of continuous relaxations of conic mixed-integer programs and, hence, improving their solvability.
An exact solution approach for portfolio optimization problems under stochastic and integer constraints
- 200 18.56 4.79 3.57 17.33 7.73 0.03 15.50 9.50 2.00 600 49.60 8.33 2.22 42.32 9.73 0.03 34.46 10.39 2.00 1000 96.15 10.19 2.38 94.93 12.97 0.03 90.25 15.38 2.00 20 200 34.05 9.06 3.11 27.11 10.97 1.10 21.23 12.00 2.00 600 96.98 9.51 4.22 79.78 12.00 1.10
, 2009
"... In this paper, we study extensions of the classical Markowitz ’ mean-variance portfolio opti-mization model. First, we consider that the expected asset returns are stochastic by introducing a probabilistic constraint imposing that the expected return of the constructed portfolio must exceed a prescr ..."
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Cited by 21 (3 self)
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In this paper, we study extensions of the classical Markowitz ’ mean-variance portfolio opti-mization model. First, we consider that the expected asset returns are stochastic by introducing a probabilistic constraint imposing that the expected return of the constructed portfolio must exceed a prescribed return level with a high confidence level. We study the deterministic equivalents of these models. In particular, we define under which types of probability distributions the determin-istic equivalents are second-order cone programs, and give exact or approximate closed-form for-mulations. Second, we account for real-world trading constraints, such as the need to diversify the investments in a number of industrial sectors, the non-profitability of holding small positions and the constraint of buying stocks by lots, modeled with integer variables. To solve the resulting problems, we propose an exact solution approach in which the uncertainty in the estimate of the expected re-turns and the integer trading restrictions are simultaneously considered. The proposed algorithmic approach rests on a non-linear branch-and-bound algorithm which features two new branching rules. The first one is a static rule, called idiosyncratic risk branching, while the second one is dynamic and called portfolio risk branching. The proposed branching rules are implemented and tested using the open-source framework of the solver Bonmin. The comparison of the computational results obtained with standard MINLP solvers and with the proposed approach shows the effectiveness of this latter which permits to solve to optimality problems with up to 200 assets in a reasonable amount of time.
Mixed Integer Second Order Cone Programming
, 2009
"... This paper deals with solving strategies for mixed integer second order cone problems. We present different lift-and-project based linear and convex quadratic cut generation techniques for mixed 0-1 second-order cone problems and present a new convergent outer approximation based approach to solve ..."
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Cited by 19 (1 self)
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This paper deals with solving strategies for mixed integer second order cone problems. We present different lift-and-project based linear and convex quadratic cut generation techniques for mixed 0-1 second-order cone problems and present a new convergent outer approximation based approach to solve mixed integer SOCPs. The latter is an extension of outer approximation based approaches for continuously differ-entiable problems to subdifferentiable second order cone constraint functions. We give numerical results for some application problems, where the cuts are applied in the context of a nonlinear branch-and-cut method and the branch-and-bound based outer approximation algorithm. The different approaches are compared to each other.
Lifting for conic mixed-integer programming
, 2011
"... Lifting is a procedure for deriving valid inequalities for mixed-integer sets from valid inequalities for suitable restrictions of those sets. Lifting has been shown to be very effective in developing strong valid inequalities for linear integer programming and it has been successfully used to sol ..."
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Cited by 15 (4 self)
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Lifting is a procedure for deriving valid inequalities for mixed-integer sets from valid inequalities for suitable restrictions of those sets. Lifting has been shown to be very effective in developing strong valid inequalities for linear integer programming and it has been successfully used to solve such problems with branch-and-cut algorithms. Here we generalize the theory of lifting to conic integer programming, i.e., integer programs with conic constraints. We show how to derive conic valid inequalities for a conic integer program from conic inequalities valid for its lowerdimensional restrictions. In order to simplify the computations, we also discuss sequence-independent lifting for conic integer programs. When the cones are restricted to nonnegative orthants, conic lifting reduces to the lifting for linear integer programming as one may expect.
A computational comparison of reformulations of the perspective relaxation: SOCP vs. cutting planes
, 2009
"... The Perspective Reformulation generates tight approximations to MINLP problems with semicontinuous variables. It can be implemented either as a Second-Order Cone Program, or as a Semi-Infinite Linear Program. We compare the two reformulations on two MIQPs in the context of exact or approximate Branc ..."
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Cited by 15 (7 self)
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The Perspective Reformulation generates tight approximations to MINLP problems with semicontinuous variables. It can be implemented either as a Second-Order Cone Program, or as a Semi-Infinite Linear Program. We compare the two reformulations on two MIQPs in the context of exact or approximate Branch-and-Cut algorithms.
Linear relaxations for transmission system planning
- IEEE Transactions on Power Systems
, 2011
"... We apply a linear relaxation procedure for polynomial optimization problems to transmission system planning. The approach recovers and improves upon existing linear models based on the DC approx-imation. We then consider the full AC problem, and obtain new linear models with nearly the same efficien ..."
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Cited by 7 (1 self)
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We apply a linear relaxation procedure for polynomial optimization problems to transmission system planning. The approach recovers and improves upon existing linear models based on the DC approx-imation. We then consider the full AC problem, and obtain new linear models with nearly the same efficiency as the linear DC models. The new models are applied to standard test systems, and produce high quality approximate solutions in reasonable computation time. 1
POLYMATROIDS AND MEAN-RISK MINIMIZATION IN DISCRETE OPTIMIZATION
, 2007
"... Abstract. In financial markets high levels of risk are associated with large returns as well as large losses, whereas with lower levels of risk, the potential for either return or loss is small. Therefore, risk management is fundamentally concerned with finding an optimal tradeoff between risk and r ..."
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Cited by 6 (0 self)
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Abstract. In financial markets high levels of risk are associated with large returns as well as large losses, whereas with lower levels of risk, the potential for either return or loss is small. Therefore, risk management is fundamentally concerned with finding an optimal tradeoff between risk and return matching an investor’s risk tolerance. Managing risk is studied mostly in a financial context; nevertheless, it is certainly relevant in any area with a significant source of uncertainty. The mean-risk tradeoff is well-studied for problems with a convex feasible set. However, this is not the case in the discrete setting, even though, in practice, portfolios are often restricted to discrete choices. In this paper we study mean-risk minimization for problems with discrete decision variables. In particular, we consider discrete optimization problems with a submodular mean-risk minimization objective. We show the connection between extended polymatroids and the convex lower envelope of this mean-risk objective. For 0-1 problems a complete linear characterization of the convex lower envelope is given. For mixed 0-1 problems we derive an exponential class of conic quadratic inequalities that are separable with the greedy algorithm.
Extending a CIP framework to solve MIQCPs
, 2010
"... This paper discusses how to build a solver for mixed integer quadratically constrained programs (MIQCPs) by extending a framework for constraint integer programming (CIP). The advantage of this approach is that we can utilize the full power of advanced MILP and CP technologies, in particular for th ..."
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Cited by 6 (2 self)
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This paper discusses how to build a solver for mixed integer quadratically constrained programs (MIQCPs) by extending a framework for constraint integer programming (CIP). The advantage of this approach is that we can utilize the full power of advanced MILP and CP technologies, in particular for the linear relaxation and the discrete components of the problem. We use an outer approximation generated by linearization of convex constraints and linear underestimation of nonconvex constraints to relax the problem. Further, we give an overview of the reformulation, separation, and propagation techniques that are used to handle the quadratic constraints efficiently. We implemented these methods in the branch-cut-and-price framework SCIP. Computational experiments indicating the potential of the approach and evaluating the impact of the algorithmic components are provided.
