A framework for solving mixed-integer semidefinite programs

Gally, Tristan; Pfetsch, Marc E.; Ulbrich, Stefan

doi:10.1080/10556788.2017.1322081

Cited by 68 publications

(60 citation statements)

References 44 publications

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“…The proof of Theorem 1 is given in Appendix C. This theorem allows the SSASP to be solved as a MI-SDP, which can be handled using a variety of optimization methods such as: branch-and-bound algorithms [10,11], outer approximations [20], or cutting-plane methods [15]. The next section presents a departure from MI-SDP to an algorithm that returns optimal solutions to SSASP, without requiring L 1,2,3 .…”

Section: Ssasp As a Mi-sdpmentioning

confidence: 99%

Algorithms for joint sensor and control nodes selection in dynamic networks

et al. 2019

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The problem of placing or selecting sensors and control nodes plays a pivotal role in the operation of dynamic networks. This paper proposes optimal algorithms and heuristics to solve the simultaneous sensor and actuator selection problem in linear dynamic networks. In particular, a sufficiency condition of static output feedback stabilizability is used to obtain the minimal set of sensors and control nodes needed to stabilize an unstable network. We show the joint sensor/actuator selection and output feedback control can be written as a mixed-integer nonconvex problem. To solve this nonconvex combinatorial problem, three methods based on (1) mixed-integer nonlinear programming, (2) binary search algorithms, and (3) simple heuristics are proposed. The first method yields optimal solutions to the selection problem-given that some constants are appropriately selected. The second method requires a database of binary sensor/actuator combinations, returns optimal solutions, and necessitates no tuning parameters. The third approach is a heuristic that yields suboptimal solutions but is computationally attractive. The theoretical properties of these methods are discussed and numerical tests on dynamic networks showcase the trade-off between optimality and computational time.

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Section: Ssasp As a Mi-sdpmentioning

confidence: 99%

Algorithms for joint sensor and control nodes selection in dynamic networks

et al. 2019

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“…Problem-specific MI-SDP strategies have been developed for problems such as binary quadratic programming [33], robust truss topology [63] or the max-cut problem [51]. More recently, rounding and Gomory cuts [12,1], branch-and-bound [29] and outer-approximation schemes [43] have also been developed, in an attempt to provide the same level of general-purpose solvers for MI-SDP as there are for mixed-integer linear optimization. Our approach is similar to the outer-approximation procedure described by [43] but leverages the specific dependency between the binary and continuous variables in our problem.…”

Section: Current Approachesmentioning

confidence: 99%

Certifiably optimal sparse inverse covariance estimation

Bertsimas¹,

Lamperski²,

Pauphilet³

2019

Math. Program.

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We consider the maximum likelihood estimation of sparse inverse covariance matrices. We demonstrate that current heuristic approaches primarily encourage robustness, instead of the desired sparsity.We give a novel approach that solves the cardinality constrained likelihood problem to certifiable optimality. The approach uses techniques from mixed-integer optimization and convex optimization, and provides a high-quality solution with a guarantee on its suboptimality, even if the algorithm is terminated early. Using a variety of synthetic and real datasets, we demonstrate that our approach can solve problems where the dimension of the inverse covariance matrix is up to 1, 000s. We also demonstrate that our approach produces significantly sparser solutions than Glasso and other popular learning procedures, makes less false discoveries, while still maintaining state-of-the-art accuracy.

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“…In fact, no approach based on straightforward polyhedral approximation can succeed. Very recently, Gally et al [20] have studied conditions in the context of mixedinteger semidefinite optimization which ensure that strong duality holds when integer values are fixed.…”

Section: Lemmamentioning

confidence: 99%

Polyhedral approximation in mixed-integer convex optimization

et al. 2017

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Generalizing both mixed-integer linear optimization and convex optimization, mixed-integer convex optimization possesses broad modeling power but has seen relatively few advances in general-purpose solvers in recent years. In this paper, we intend to provide a broadly accessible introduction to our recent work in developing algorithms and software for this problem class. Our approach is based on constructing polyhedral outer approximations of the convex constraints, resulting in a global solution by solving a finite number of mixed-integer linear and continuous convex subproblems. The key advance we present is to strengthen the polyhedral approximations by constructing them in a higher-dimensional space. In order to automate this extended formulation we rely on the algebraic modeling technique of disciplined convex programming (DCP), and for generality and ease of implementation we use conic representations of the convex constraints. Although our framework requires a manual translation of existing models into DCP form, after performing this transformation on the MINLPLIB2 benchmark library we were able to solve a number of unsolved instances and on many other instances achieve superior performance compared with state-of-the-art solvers like Bonmin, SCIP, and Artelys Knitro

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A framework for solving mixed-integer semidefinite programs

Cited by 68 publications

References 44 publications

Algorithms for joint sensor and control nodes selection in dynamic networks

Algorithms for joint sensor and control nodes selection in dynamic networks

Certifiably optimal sparse inverse covariance estimation

Polyhedral approximation in mixed-integer convex optimization

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