Ideal formulations for constrained convex optimization problems with indicator variables

Wei, Linchuan; Gómez, Andrés; Küçükyavuz, Simge

doi:10.1007/s10107-021-01734-y

Cited by 18 publications

(4 citation statements)

References 54 publications

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“…In [2], the authors consider the set S 2 with the additional constraint x ≤ 1, and obtain an extended formulation for the convex hull of this set that is SDP representable. More results regarding the convexification of Problem (MIQPI) can be found in [11,13,21,20].…”

Section: And Derive Conv(z −mentioning

confidence: 99%

Explicit convex hull description of bivariate quadratic sets with indicator variables

Rosa¹,

Khajavirad²

2022

Preprint

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We consider the nonconvex set S n = {(x, X, z) :which is closely related to the feasible region of several difficult nonconvex optimization problems such as the best subset selection and constrained portfolio optimization. Utilizing ideas from convex analysis and disjunctive programming, we obtain an explicit description for the closure of the convex hull of S 2 in the space of original variables. In order to generate valid inequalities corresponding to supporting hyperplanes of the convex hull of S 2 , we present a simple separation algorithm that can be incorporated in branch-and-cut based solvers to enhance the quality of existing relaxations.

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Section: And Derive Conv(z −mentioning

confidence: 99%

Explicit convex hull description of bivariate quadratic sets with indicator variables

Rosa¹,

Khajavirad²

2022

Preprint

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“…Atamtürk and Gómez [7] give the convex hull description of a rank-one function with free continuous variables, and propose an SDP formulation to tackle quadratic optimization problems with free variables arising in sparse regression. Wei et al [50,51] extend those results, deriving ideal formulations for rank-one functions with arbitrary constraints on the indicator variables x. These formulations are shown to be effective in sparse regression problems; however as they do not account for the non-negativity constraints on the continuous variables, they are weak for (1).…”

Section: Introductionmentioning

confidence: 97%

Supermodularity and valid inequalities for quadratic optimization with indicators

Atamtürk¹,

Gómez²

2020

Preprint

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We study the minimization of a rank-one quadratic with indicators and show that the underlying set function obtained by projecting out the continuous variables is supermodular. Although supermodular minimization is, in general, difficult, the specific set function for the rank-one quadratic can be minimized in linear time. We show that the convex hull of the epigraph of the quadratic can be obtaining from inequalities for the underlying supermodular set function by lifting them into nonlinear inequalities in the original space of variables. Explicit forms of the convex-hull description are given, both in the original space of variables and in an extended formulation via conic quadraticrepresentable inequalities, along with a polynomial separation algorithm. Computational experiments indicate that the lifted supermodular inequalities in conic quadratic form are quite effective in reducing the integrality gap for quadratic optimization with indicators.

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“…Atamtürk and Gómez [7] give the convex hull description of a rank-one function with free continuous variables, and propose an SDP formulation to tackle quadratic optimization problems with free variables arising in sparse regression. Wei et al [51,52] extend those results, deriving ideal formulations for rank-one functions with arbitrary constraints on the indicator variables x. These formulations are shown to be effective in sparse regression problems; however as they do not account for the non-negativity constraints on the continuous variables, they are weak for (1).…”

Section: Introductionmentioning

confidence: 97%

Supermodularity and valid inequalities for quadratic optimization with indicators

Atamtürk

Gómez

2022

Math. Program.

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We study the minimization of a rank-one quadratic with indicators and show that the underlying set function obtained by projecting out the continuous variables is supermodular. Although supermodular minimization is, in general, difficult, the specific set function for the rank-one quadratic can be minimized in linear time. We show that the convex hull of the epigraph of the quadratic can be obtained from inequalities for the underlying supermodular set function by lifting them into nonlinear inequalities in the original space of variables. Explicit forms of the convex-hull description are given, both in the original space of variables and in an extended formulation via conic quadratic-representable inequalities, along with a polynomial separation algorithm. Computational experiments indicate that the lifted supermodular inequalities in conic quadratic form are quite effective in reducing the integrality gap for quadratic optimization with indicators.

show abstract

Ideal formulations for constrained convex optimization problems with indicator variables

Cited by 18 publications

References 54 publications

Explicit convex hull description of bivariate quadratic sets with indicator variables

Explicit convex hull description of bivariate quadratic sets with indicator variables

Supermodularity and valid inequalities for quadratic optimization with indicators

Supermodularity and valid inequalities for quadratic optimization with indicators

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