Cardinality Minimization, Constraints, and Regularization: A Survey

Tillmann, Andreas M.; Bienstock, Daniel; Lodi, Andrea; Schwartz, Alexandra

doi:10.48550/arxiv.2106.09606

Cited by 9 publications

(15 citation statements)

References 298 publications

(476 reference statements)

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“…Assume that point x * = (0, 0, 1) is feasible for system (38). According to Definition 3.4 (7), we say that RCPLD holds at…”

Section: Rcpld As a Sufficient Condition For Mscqmentioning

confidence: 99%

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Optimality conditions and constraint qualifications for cardinality constrained optimization problems

Xiao¹,

Ye²

2022

Preprint

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The cardinality constrained optimization problem (CCOP) is an optimization problem where the maximum number of nonzero components of any feasible point is bounded. In this paper, we consider CCOP as a mathematical program with disjunctive subspaces constraints (MPDSC). Since a subspace is a special case of a convex polyhedral set, MPDSC is a special case of the mathematical program with disjunctive constraints (MPDC). Using the special structure of subspaces, we are able to obtain more precise formulas for the tangent and (directional) normal cones for the disjunctive set of subspaces. We then obtain first and second order optimality conditions by using the corresponding results from MPDC. Thanks to the special structure of the subspace, we are able to obtain some results for MPDSC that do not hold in general for MPDC. In particular we show that the relaxed constant positive linear dependence (RCPLD) is a sufficient condition for the metric subregularity/error bound property for MPDSC which is not true for MPDC in general. Finally we show that under all constraint qualifications presented in this paper, certain exact penalization holds for CCOP.

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“…Assume that point x * = (0, 0, 1) is feasible for system (38). According to Definition 3.4 (7), we say that RCPLD holds at…”

Section: Rcpld As a Sufficient Condition For Mscqmentioning

confidence: 99%

“…∈ S 3 , we only have two possible subsystems. Therefore, the piecewise RCPLD holds for system (38) at x * if RCPLD holds for each of the following two subsystems at x * :…”

Section: And Any Set Of Linearly Independent Vectorsmentioning

confidence: 99%

Optimality conditions and constraint qualifications for cardinality constrained optimization problems

Xiao¹,

Ye²

2022

Preprint

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“…As a result, the constraints impose a non-convex feasible domain on the problem. Non-convex optimization problems as such are of practical significance in regression, machine learning, and signal processing [28].…”

Section: Mixed-integer Programmingmentioning

confidence: 99%

“…Finally, in Section VI, we demonstrate the application of our method in solving MIP problems, using an example instance of a sparse optimization problem. Such cardinality-constrained optimization problems are also NP-hard [27] and of practical importance in compressed sensing, signal processing, and computational finance [28].…”

Section: Introductionmentioning

confidence: 99%

Mixed-Integer Programming Using a Bosonic Quantum Computer

Khosravi¹,

Scherer²,

Ronagh³

2021

Preprint

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We propose a scheme for solving mixed-integer programming problems in which the optimization problem is translated to a ground-state preparation problem on a set of bosonic quantum field modes (qumodes). We perform numerical demonstrations by simulating a circuit-based optical quantum computer with each individual qumode prepared in a Gaussian state. We simulate an adiabatic evolution from an initial mixing Hamiltonian, written in terms of the momentum operators of the qumodes, to a final Hamiltonian which is a polynomial of the position and boson number operators. In these demonstrations, we solve a variety of small non-convex optimization problems in integer programming, continuous non-convex optimization, and mixed-integer programming.

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“…In recent years, Sparse Convex Optimization (SCO) has gained considerable attention in several disciplines, from machine learning and engineering to economics and finance [4,6,39]. Several mathematical optimization problems in this context can be formulated as a general convex optimization problem subject to a constraint that allows only up to a certain number of decision variables to be nonzero.…”

Section: Introductionmentioning

confidence: 99%

Sparse Convex Optimization Toolkit: A Mixed-Integer Framework

Olama¹,

Camponogara²,

Kronqvist³

2022

Preprint

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This paper proposes an open-source distributed solver for solving Sparse Convex Optimization (SCO) problems over computational networks. Motivated by past algorithmic advances in mixed-integer optimization, the Sparse Convex Optimization Toolkit (SCOT) adopts a mixed-integer approach to find exact solutions to SCO problems. In particular, SCOT brings together various techniques to transform the original SCO problem into an equivalent convex Mixed-Integer Nonlinear Programming (MINLP) problem that can benefit from high-performance and parallel computing platforms. To solve the equivalent mixed-integer problem, we present the Distributed Hybrid Outer Approximation (DiHOA) algorithm that builds upon the LP/NLP based branch-and-bound and is tailored for this specific problem structure. The DiHOA algorithm combines the so-called single-and multi-tree outer approximation, naturally integrates a decentralized algorithm for distributed convex nonlinear subproblems, and utilizes enhancement techniques such as quadratic cuts. Finally, we present detailed computational experiments that show the benefit of our solver through numerical benchmarks on 140 SCO problems with distributed datasets. To show the overall efficiency of SCOT we also provide performance profiles comparing SCOT to other state-of-the-art MINLP solvers.

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Cardinality Minimization, Constraints, and Regularization: A Survey

Cited by 9 publications

References 298 publications

Optimality conditions and constraint qualifications for cardinality constrained optimization problems

Optimality conditions and constraint qualifications for cardinality constrained optimization problems

Mixed-Integer Programming Using a Bosonic Quantum Computer

Sparse Convex Optimization Toolkit: A Mixed-Integer Framework

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