Tractable ADMM schemes for computing KKT points and local minimizers for $$\ell _0$$-minimization problems

Xie, Yue; Shanbhag, Uday V.

doi:10.1007/s10589-020-00227-6

Cited by 8 publications

(5 citation statements)

References 32 publications

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“…3.2 . Both regimes (significantly different as noted, e.g., by [ 36 ]) are met in the case of sparse linear regression treated in Sect. 3.3 .…”

Section: Introductionmentioning

confidence: 76%

“…Interestingly enough, most of the papers using purely continuous reformulations of problems involving in an explicit way, either in the objective or in the constraint, focus on local solutions via NLP solvers, with considerable success, demonstrated by convergence results and impressive empirical findings; see, e.g. [ 14 , 22 , 25 , 36 ].…”

Section: Introductionmentioning

confidence: 99%

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Conic formulation of QPCCs applied to truly sparse QPs

Bomze

Peng

2022

Comput Optim Appl

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We study (nonconvex) quadratic optimization problems with complementarity constraints, establishing an exact completely positive reformulation under—apparently new—mild conditions involving only the constraints, not the objective. Moreover, we also give the conditions for strong conic duality between the obtained completely positive problem and its dual. Our approach is based on purely continuous models which avoid any branching or use of large constants in implementation. An application to pursuing interpretable sparse solutions of quadratic optimization problems is shown to satisfy our settings, and therefore we link quadratic problems with an exact sparsity term $$\Vert {{\mathsf {x}}}\Vert _0$$ ‖ x ‖ 0 to copositive optimization. The covered problem class includes sparse least-squares regression under linear constraints, for instance. Numerical comparisons between our method and other approximations are reported from the perspective of the objective function value.

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“…3.2 . Both regimes (significantly different as noted, e.g., by [ 36 ]) are met in the case of sparse linear regression treated in Sect. 3.3 .…”

Section: Introductionmentioning

confidence: 76%

Section: Introductionmentioning

confidence: 99%

Conic formulation of QPCCs applied to truly sparse QPs

Bomze

Peng

2022

Comput Optim Appl

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“…An advantage of using a surrogate, as opposed to directly solving a discrete optimization problem, is computational efficiency. Many existing methods involving the 0 -function, e.g., best subset selection and cardinality constraint, introduce auxiliary binary integer variables [4,14] or reformulate the problem using complementarity constraints [6,12,34]. The potential burden of increased computational time from such reformulations could be addressed by using continuous functions instead, and we will demonstrate the effectiveness of the proposed approach in the § 4.…”

Section: Difference-of-convex Approximationmentioning

confidence: 94%

A Cardinality Minimization Approach to Security-Constrained Economic Dispatch

Troxell¹,

Ahn²,

Gangammanavar³

2021

Preprint

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We present a threshold-based cardinality minimization formulation to model the securityconstrained economic dispatch problem. The model aims to minimize the operating cost of the system while simultaneously reducing the number of lines operating in emergency operating zones during contingency events. The model allows the system operator to monitor the duration for which lines operate in emergency zones and ensure that they are within the acceptable reliability standards determined by the system operators. We develop a continuous difference-of-convex approximation of the cardinality minimization problem and a solution method to solve the problem. Our numerical experiments demonstrate that the cardinality minimization approach reduces the overall system operating cost as well as avoids prolonged periods of high electricity prices during contingency events.

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“…In [333], an ADMM was designed for the relaxed reformulation of the cardinality regularization problem 0 -reg(ρ, Ax ≥ b), and the authors of [343] use the observation that…”

Section: Other Relaxations Of Cardinality Constraintsmentioning

confidence: 99%

“…Nevertheless, in case f is convex and thus f N ≡ 0, solving the minimization problem with regards to (v , w ) reduces to projecting (x, y ) onto the set of points (v , w ) with v i (1 − w i ) = 0 for all i ∈ [n], for which a closed-form solution is available. A closed-form solution for a nonconvex, but quadratic function f N is given in [333]. Moreover, recall that ADMM schemes are also popular for convex 1 -based models, cf.…”

Section: Other Relaxations Of Cardinality Constraintsmentioning

confidence: 99%

Cardinality Minimization, Constraints, and Regularization: A Survey

Tillmann¹,

Bienstock²,

Lodi³

et al. 2021

Preprint

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We survey optimization problems that involve the cardinality of variable vectors in constraints or the objective function. We provide a unified viewpoint on the general problem classes and models, and give concrete examples from diverse application fields such as signal and image processing, portfolio selection, or machine learning. The paper discusses general-purpose modeling techniques and broadly applicable as well as problem-specific exact and heuristic solution approaches. While our perspective is that of mathematical optimization, a main goal of this work is to reach out to and build bridges between the different communities in which cardinality optimization problems are frequently encountered. In particular, we highlight that modern mixed-integer programming, which is often regarded as impractical due to commonly unsatisfactory behavior of black-box solvers applied to generic problem formulations, can in fact produce provably high-quality or even optimal solutions for cardinality optimization problems, even in large-scale real-world settings. Achieving such performance typically draws on the merits of problem-specific knowledge that may stem from different fields of application and, e.g., shed light on structural properties of a model or its solutions, or lead to the development of efficient heuristics; we also provide some illustrative examples.

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Tractable ADMM schemes for computing KKT points and local minimizers for $$\ell _0$$-minimization problems

Cited by 8 publications

References 32 publications

Conic formulation of QPCCs applied to truly sparse QPs

Conic formulation of QPCCs applied to truly sparse QPs

A Cardinality Minimization Approach to Security-Constrained Economic Dispatch

Cardinality Minimization, Constraints, and Regularization: A Survey

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