Semidefinite Programming Reformulation Research Articles

Picking winners: Diversification through portfolio optimization

We develop a general framework for selecting a small pool of candidate solutions to maximize the chances that one will be optimal for a combinatorial optimization problem, under a linear and additive random payoff function. We formulate this problem using a two‐stage distributionally robust model, with a mixed 0–1 semidefinite program. This approach allows us to exploit the “diversification” effect inherent in the problem to address how different candidate solutions can be selected to improve the chances that one will attain a high ex post payoff. More interestingly, using this distributionally robust optimization approach, our model recovers the “evil twin” strategy, well known in the field of football pool betting, under appropriate settings. We also address the computational challenges of scaling up our approach to construct a moderate number of candidate solutions to increase the chances of finding one that performs well. To this end, we develop a sequential optimization approach based on a compact semidefinite programming reformulation of the problem. Extensive numerical results show the superiority of our approach over existing methods.

Two-Stage Co-Optimization for Utility-Social Systems With Social-Aware P2P Trading

Effective utility system management is fundamental and critical for ensuring the normal activities, operations, and services in cities and urban areas. In that regard, the advanced information and communication technologies underpinning smart cities enable close linkages and coordination of different subutility systems, which is now attracting research attention. To increase operational efficiency, we propose a two-stage optimal co-management model for an integrated urban utility system comprised of water, power, gas, and heating systems, namely, integrated water-energy hubs (IWEHs). The proposed IWEH facilitates coordination between multienergy and water sectors via close energy conversion and can enhance the operational efficiency of an integrated urban utility system. In particular, we incorporate social-aware peer-to-peer (P2P) resource trading in the optimization model, in which operators of an IWEH can trade energy and water with other interconnected IWEHs. To cope with renewable generation and load uncertainties and mitigate their negative impacts, a two-stage distributionally robust optimization (DRO) is developed to capture the uncertainties, using a semidefinite programming reformulation. To demonstrate our model’s effectiveness and practical values, we design representative case studies that simulate four interconnected IWEH communities. The results show that DRO is more effective than robust optimization (RO) and stochastic optimization (SO) for avoiding excessive conservativeness and rendering practical utilities, without requiring enormous data samples. This work reveals a desirable methodological approach to optimize the water–energy–social nexus for increased economic and system-usage efficiency for the entire (integrated) urban utility system. Furthermore, the proposed model incorporates social participations by citizens to engage in urban utility management for increased operation efficiency of cities and urban areas.

Decision Rule Approaches for Pessimistic Bilevel Linear Programs Under Moment Ambiguity with Facility Location Applications

We study a pessimistic stochastic bilevel program in the context of sequential two-player games, where the leader makes a binary here-and-now decision, and the follower responds with a continuous wait-and-see decision after observing the leader’s action and revelation of uncertainty. We assume that only the information regarding the mean, covariance, and support is known. We formulate the problem as a distributionally robust (DR) two-stage problem. The pessimistic DR bilevel program is shown to be equivalent to a generic two-stage distributionally robust stochastic (nonlinear) program with both a random objective and random constraints under proper conditions of ambiguity sets. Under continuous distributions, using linear decision rule approaches, we construct upper bounds on the pessimistic DR bilevel program based on (1) a 0-1 semidefinite programming (SDP) approximation and (2) an exact 0-1 copositive programming reformulation. When the ambiguity set is restricted to discrete distributions, an exact 0-1 SDP reformulation is developed, and explicit construction of the worst-case distribution is derived. To further improve the computation of the proposed 0-1 SDPs, a cutting-plane framework is developed. Moreover, based on a mixed-integer linear programming approximation, another cutting-plane algorithm is proposed. Extensive numerical studies are conducted to demonstrate the effectiveness of the proposed approaches on a facility location problem. History: Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0168 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0168 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

A worst-case approach to topology optimization for maximum stiffness under uncertain boundary displacement

We present a worst-case approach to topology optimization (TO) for maximum stiffness under boundary displacement parametrized by a matrix-valued scaling function times an uncertain vector giving its direction. The objective function in the TO problem is the minimum of the potential energy maximized over the set of boundary displacements, which in the absence of prescribed loads means maximizing the reaction loads arising from enforcing the boundary displacement. It is shown that the TO problem can be cast as the minimization of the maximum eigenvalue of a matrix depending on solutions to a small number of (linear elastic) state problems. Numerical solution of this potentially non-smooth problem using algorithms for smooth optimization, a non-linear semi-definite programming reformulation, and a non-smooth bundle method is discussed and tested.

Exploiting partial correlations in distributionally robust optimization

In this paper, we identify partial correlation information structures that allow for simpler reformulations in evaluating the maximum expected value of mixed integer linear programs with random objective coefficients. To this end, assuming only the knowledge of the mean and the covariance matrix entries restricted to block-diagonal patterns, we develop a reduced semidefinite programming formulation, the complexity of solving which is related to characterizing a suitable projection of the convex hull of the set $$\{(\mathbf x , \mathbf x {} \mathbf x '): \mathbf x \in \mathcal {X}\}$$ where $$\mathcal {X}$$ is the feasible region. In some cases, this lends itself to efficient representations that result in polynomial-time solvable instances, most notably for the distributionally robust appointment scheduling problem with random job durations as well as for computing tight bounds in the newsvendor problem, project evaluation and review technique networks and linear assignment problems. To the best of our knowledge, this is the first example of a distributionally robust optimization formulation for appointment scheduling that permits a tight polynomial-time solvable semidefinite programming reformulation which explicitly captures partially known correlation information between uncertain processing times of the jobs to be scheduled. We also discuss extensions where the random coefficients are assumed to be non-negative and additional overlapping correlation information is available.

Sum-of-Squares Optimization without Semidefinite Programming

We propose a homogeneous primal-dual interior-point method to solve sum-of-squares optimization problems by combining nonsymmetric conic optimization techniques and polynomial interpolation. The approach optimizes directly over the sum-of-squares cone and its dual, circumventing the semidefinite programming (SDP) reformulation, which requires a large number of auxiliary variables when the degree of sum-of-squares polynomials is large. As a result, it has substantially lower theoretical time and space complexity than the conventional SDP-based approach as the degree increases. Although our approach avoids the SDP reformulation, an optimal solution to the semidefinite program can be recovered with little additional effort. Computational results confirm that the proposed method is several orders of magnitude faster than the SDP-based approach for optimization problems over high-degree sum-of-squares polynomials.

Exploiting Partial Correlations in Distributionally Robust Optimization

In this paper, we identify partial correlation information structures that allow for simpler reformulations in evaluating the maximum expected value of mixed integer linear programs with random objective coefficients. To this end, assuming only the knowledge of the mean and the covariance matrix entries restricted to block-diagonal patterns, we develop a reduced semidefinite programming formulation, the complexity of solving which is related to characterizing a suitable projection of the convex hull of the set {(x, xx' ) : x ∈ X } where X is the feasible region. In some cases, this lends itself to efficient representations that result in polynomial-time solvable instances, most notably for the distributionally robust appointment scheduling problem with random job durations as well as for computing tight bounds in Project Evaluation and Review Technique (PERT) networks and linear assignment problems. To the best of our knowledge, this is the first example of a distributionally robust optimization formulation for appointment scheduling that permits a tight polynomial-time solvable semidefinite programming reformulation which explicitly captures partially known correlation information between uncertain processing times of the jobs to be scheduled.

A Distributionally Robust Linear Receiver Design for Multi-Access Space-Time Block Coded MIMO Systems

A receiver design problem for multi-access space-time block coded multiple-input multiple-output systems is considered. To hedge the mismatch between the true and the estimated channel state information (CSI), several robust receivers have been developed in the past decades. Among these receivers, the Gaussian robust receiver has been shown to be superior in performance. This receiver is designed based on the assumption that the CSI mismatch has Gaussian distribution. However, in real-world applications, the assumption of Guassianity might not hold. Motivated by this fact, a more general distributionally robust receiver is proposed in this paper, where only the mean and the variance of the CSI mismatch distribution are required in the receiver design. A tractable semi-definite programming (SDP) reformulation of the robust receiver design is developed. To suppress the self-interferences, a more advanced distributionally robust receiver is proposed. A tight convex approximation is given and the corresponding tractable SDP reformulation is developed. Moreover, for the sake of easy implementation, we present a simplified distributionally robust receiver. Simulations results are provided to show the effectiveness of our design by comparing with some existing well-known receivers.

Exponential Lower Bounds for Polytopes in Combinatorial Optimization

We solve a 20-year old problem posed by Yannakakis and prove that no polynomial-size linear program (LP) exists whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope and the stable set polytope. These results were discovered through a new connection that we make between one-way quantum communication protocols and semidefinite programming reformulations of LPs.

Distributionally Robust Stochastic Knapsack Problem

This paper considers a distributionally robust version of a quadratic knapsack problem. In this model, a subsets of items is selected to maximizes the total profit while requiring that a set of knapsack constraints be satisfied with high probability. In contrast to the stochastic programming version of this problem, we assume that only part of the information on random data is known, i.e., the first and second moment of the random variables, their joint support, and possibly an independence assumption. As for the binary constraints, special interest is given to the corresponding semidefinite programming (SDP) relaxation. While in the case that the model only has a single knapsack constraint we present an SDP reformulation for this relaxation, the case of multiple knapsack constraints is more challenging. Instead, two tractable methods are presented for providing upper and lower bounds (with its associated conservative solution) on the SDP relaxation. An extensive computational study is given to illustrate the tightness of these bounds and the value of the proposed distributionally robust approach.

Quadratic Optimization over a Second-Order Cone with Linear Equality Constraints

This paper studies the nonhomogeneous quadratic programming problem over a second-order cone with linear equality constraints. When the feasible region is bounded, we show that an optimal solution of the problem can be found in polynomial time. When the feasible region is unbounded, a semidefinite programming (SDP) reformulation is constructed to find the optimal objective value of the original problem in polynomial time. In addition, we provide two sufficient conditions, under which, if the optimal objective value is finite, we show the optimal solution of SDP reformulation can be decomposed into the original space to generate an optimal solution of the original problem in polynomial time. Otherwise, a recession direction can be identified in polynomial time. Numerical examples are included to illustrate the effectiveness of the proposed approach.

On minimizing the ratio of quadratic functions over an ellipsoid

We study the optimization problem (RQ) of minimizing the ratio of two quadratic functions over a possibly degenerate ellipsoid. The well definition of problem (RQ) is fully characterized. We show any well-defined (RQ) admits a semidefinite programming reformulation (SDP) without any assumption. Moreover, the minimum of (RQ) is attained if and only if (SDP) has a unique solution.

Complexity and nonlinear semidefinite programming reformulation of $$\ell _1$$ ℓ 1 -constrained nonconvex quadratic optimization

In this article, we show that the \(\ell _1\)-constrained nonconvex quadratic optimization problem (QPL1) is NP-hard, even when the off-diagonal elements are all nonnegative. Then we give an answer to Pinar and Teboulle’s open problem on the nonlinear semidefinite programming relaxation of (QPL1). The analytical approach is extended to \(\ell _p\)-constrained quadratic programs with \(1<p<2\).

Models and algorithms for distributionally robust least squares problems

We present three different robust frameworks using probabilistic ambiguity descriptions of the data in least squares problems. These probability ambiguity descriptions are given by: (1) confidence region over the first two moments; (2) bounds on the probability measure with moments constraints; (3) the Kantorovich probability distance from a given measure. For the first case, we give an equivalent formulation and show that the optimization problem can be solved using a semidefinite optimization reformulation or polynomial time algorithms. For the second case, we derive the equivalent Lagrangian problem and show that it is a convex stochastic programming problem. We further analyze three special subcases: (i) finite support; (ii) measure bounds by a reference probability measure; (iii) measure bounds by two reference probability measures with known density functions. We show that case (i) has an equivalent semidefinite programming reformulation and the sample average approximations of case (ii) and (iii) have equivalent semidefinite programming reformulations. For ambiguity description (3), we show that the finite support case can be solved by using an equivalent second order cone programming reformulation.