Algorithm for cardinality-constrained quadratic optimization

Bertsimas, Dimitris; Shioda, Romy

doi:10.1007/s10589-007-9126-9

Cited by 228 publications

(202 citation statements)

References 24 publications

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“…For given m observed data points (a i , b i ) with a i ∈ n and b i ∈ , while one always wants to minimize the least square measure of m i=1 (a T i x−b i ) 2 , he/she often wants to achieve the goal with only a subset of the prediction variables in x (see Arthanari and Dodge (1993); Bertsimas and Shioda (2009);Miller (2002)). This subset selection problem can be formu- The subset selection problem is a special case of (P) where the constraints of semi-continuous variables (3) and (4) are absent.…”

Section: Problem and Backgroundmentioning

confidence: 99%

“…The solution method in Bonami and Lejeune (2009) is a branch-and-bound method based on continuous relaxation and special branching rules. Bertsimas and Shioda (2009) presented a specialized branch-and-bound method for (P) where a convex quadratic programming relaxation at each node is solved via Lemke's method. Bienstock (1996) developed a branch-and-cut method for solving cardinality constrained quadratic programming problems using a surrogate constraint approach.…”

Section: Problem and Backgroundmentioning

confidence: 99%

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Improving the Performance of MIQP Solvers for Quadratic Programs with Cardinality and Minimum Threshold Constraints: A Semidefinite Program Approach

Zheng

Sun

2014

INFORMS Journal on Computing

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We consider in this paper quadratic programming problems with cardinality and minimum threshold constraints which arise naturally in various real-world applications such as portfolio selection and subset selection in regression. This class of problems can be formulated as mixed-integer 0-1 quadratic programs. We propose a new semidefinite program (SDP) approach for computing the "best" diagonal decomposition that gives the tightest continuous relaxation of the perspective reformulation of the problem. We also give an alternative way of deriving the perspective reformulation by applying a special Lagrangian decomposition scheme to the diagonal decomposition of the problem. This derivation can be viewed as a "dual" method to the convexification method employing the perspective function on semi-continuous variables. Computational results show that the proposed SDP approach can be advantageous for improving the performance of MIQP solvers when applied to the perspective reformulations of the problem.

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Section: Problem and Backgroundmentioning

confidence: 99%

Section: Problem and Backgroundmentioning

confidence: 99%

Improving the Performance of MIQP Solvers for Quadratic Programs with Cardinality and Minimum Threshold Constraints: A Semidefinite Program Approach

Zheng

Sun

2014

INFORMS Journal on Computing

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“…Among the first studies, Blog, Van der Hoeck, Rinnooy & Timmer (1983) propose a dynamic programming heuristic for small portfolios, and Bienstock (1996) proposes a surrogate constraint in lieu of the cardinality constraint. This approach is extended in Bertsimas & Shioda (2009), who propose a convex relaxation and pivoting method. More recently, Gao & Li (2013) develop a Lagrangian relaxation and geometric approach which exploits a special symmetric property of the quadratic objective.…”

Section: Related Literaturementioning

confidence: 99%

Large-Scale Loan Portfolio Selection

Sirignano¹,

Tsoukalas

Giesecke

2016

Operations Research

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We consider the problem of optimally selecting a large portfolio of risky loans, such as mortgages, credit cards, auto loans, student loans, or business loans. Examples include loan portfolios held by financial institutions and fixed-income investors as well as pools of loans backing mortgage-and asset-backed securities. The size of these portfolios can range from the thousands to even hundreds of thousands. Optimal portfolio selection requires the solution of a high-dimensional nonlinear integer program and is extremely computationally challenging. For larger portfolios, this optimization problem is intractable. We propose an approximate optimization approach that yields an asymptotically optimal portfolio for a broad class of data-driven models of loan delinquency and prepayment. We prove that the asymptotically optimal portfolio converges to the optimal portfolio as the portfolio size grows large. Numerical case studies using actual loan data demonstrate its computational efficiency. The asymptotically optimal portfolio's computational cost does not increase with the size of the portfolio. It is typically many orders of magnitude faster than nonlinear integer program solvers while also being highly accurate even for moderate-sized portfolios. AbstractWe consider the problem of optimally selecting a large portfolio of risky loans, such as mortgages, credit cards, auto loans, student loans, or business loans. Examples include loan portfolios held by financial institutions and fixed-income investors as well as pools of loans backing mortgage-and asset-backed securities. The size of these portfolios can range from the thousands to even hundreds of thousands. Optimal portfolio selection requires the solution of a high-dimensional nonlinear integer program and is extremely computationally challenging. For larger portfolios, this optimization problem is intractable. We propose an approximate optimization approach that yields an asymptotically optimal portfolio for a broad class of data-driven models of loan delinquency and prepayment. We prove that the asymptotically optimal portfolio converges to the optimal portfolio as the portfolio size grows large. Numerical case studies using actual loan data demonstrate its computational efficiency. The asymptotically optimal portfolio's computational cost does not increase with the size of the portfolio. It is typically many orders of magnitude faster than nonlinear integer program solvers while also being highly accurate even for moderate-sized portfolios.

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“…Although there are exact algorithms for the solution of MIQPs (see [5,6,7,25]), many researchers and portfolio managers prefer to use heuristics approaches (see [3,9,11,15,17,26]). Some of these heuristics vary among evolutionary algorithms, tabu search, and simulated annealing (see [15,26]).…”

Section: The Cardinality Constrained Markowitz Mean-variance Modelmentioning

confidence: 99%

“…This portfolio corresponds to the extreme point (of maximum cardinality) of the efficient frontier (or Pareto front) of the cardinality/mean-variance biobjective optimization problem (5). The third one is a classical Markowitz related portfolio and is obtained by minimizing variance under no short-selling min w∈R N w Qw subject to e w = 1,…”

Section: In-sample Optimizationmentioning

confidence: 99%

Efficient Cardinality/Mean-Variance Portfolios

Brito

Vicente

2014

IFIP Advances in Information and Communication Technology

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Abstract. We propose a novel approach to handle cardinality in portfolio selection, by means of a biobjective cardinality/mean-variance problem, allowing the investor to analyze the efficient tradeoff between returnrisk and number of active positions. Recent progress in multiobjective optimization without derivatives allow us to robustly compute (in-sample) the whole cardinality/mean-variance efficient frontier, for a variety of data sets and mean-variance models. Our results show that a significant number of efficient cardinality/mean-variance portfolios can overcome (out-of-sample) the naive strategy, while keeping transaction costs relatively low.

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Algorithm for cardinality-constrained quadratic optimization

Cited by 228 publications

References 24 publications

Improving the Performance of MIQP Solvers for Quadratic Programs with Cardinality and Minimum Threshold Constraints: A Semidefinite Program Approach

Improving the Performance of MIQP Solvers for Quadratic Programs with Cardinality and Minimum Threshold Constraints: A Semidefinite Program Approach

Large-Scale Loan Portfolio Selection

Efficient Cardinality/Mean-Variance Portfolios

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