Ideal formulations for constrained convex optimization problems with indicator variables

Motivated by modern regression applications, in this paper, we study the convexification of a class of convex optimization problems with indicator variables and combinatorial constraints on the indicators. Unlike most of the previous work on convexification of sparse regression problems, we simultan...

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Vydáno v:Mathematical programming Ročník 192; číslo 1-2; s. 57 - 88
Hlavní autoři: Wei, Linchuan, Gómez, Andrés, Küçükyavuz, Simge
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2022
Springer
Springer Nature B.V
Témata:
Calculus of Variations and Optimal Control; Optimization
Collinearity
Combinatorial analysis
Combinatorics
Computational geometry
Constraints
Convex analysis
Convexity
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Optimization
Regression
Theoretical
Perspective formulation
Combinatorial constraints
Convexification
Indicator variables
ISSN:0025-5610, 1436-4646
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Popis
Shrnutí:Motivated by modern regression applications, in this paper, we study the convexification of a class of convex optimization problems with indicator variables and combinatorial constraints on the indicators. Unlike most of the previous work on convexification of sparse regression problems, we simultaneously consider the nonlinear non-separable objective, indicator variables, and combinatorial constraints. Specifically, we give the convex hull description of the epigraph of the composition of a one-dimensional convex function and an affine function under arbitrary combinatorial constraints. As special cases of this result, we derive ideal convexifications for problems with hierarchy, multi-collinearity, and sparsity constraints. Moreover, we also give a short proof that for a separable objective function, the perspective reformulation is ideal independent from the constraints of the problem. Our computational experiments with sparse regression problems demonstrate the potential of the proposed approach in improving the relaxation quality without significant computational overhead.
Bibliografie:ObjectType-Article-1
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-021-01734-y