A novel method for modelling cardinality and rank constraints

Constraints on the cardinality or rank of decision variables in optimization problems are generally modelled separately from algebraic constraints. In this paper we show that cardinality constraints on vectors and rank constraints on matrices can be represented using purely algebraic constraints on continuous variables, by exploiting classical results on the Ky Fan norm for matrices and its analogous norm for vectors. Using this technique, a vector cardinality constraint can be modelled via introduction of a small number of additional variables and linear constraints, in conjunction with a single bilinear inequality. Analogously, a matrix rank constraint can be modelled via introduction of additional matrix variables and linear matrix inequalities, in conjunction with a single bilinear matrix inequality. We discuss a number of variations on cardinality and rank constraints that can be modelled in a similar way.

[1]  Yazid M. Sharaiha,et al.  Heuristics for cardinality constrained portfolio optimisation , 2000, Comput. Oper. Res..

[2]  Duan Li,et al.  A polynomial case of the cardinality-constrained quadratic optimization problem , 2013, J. Glob. Optim..

[3]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[4]  Daniel Pérez Palomar,et al.  Rank-Constrained Separable Semidefinite Programming With Applications to Optimal Beamforming , 2010, IEEE Transactions on Signal Processing.

[5]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[6]  Masakazu Kojima,et al.  Exact Solutions of Some Nonconvex Quadratic Optimization Problems via SDP and SOCP Relaxations , 2003, Comput. Optim. Appl..

[7]  Michael L. Overton,et al.  Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices , 2015, Math. Program..

[8]  A. Atkinson.  Subset Selection in Regression , 1992 .

[9]  H. Zou,et al.  Regularization and variable selection via the elastic net , 2005 .

[10]  Dimitris Bertsimas,et al.  Algorithm for cardinality-constrained quadratic optimization , 2009, Comput. Optim. Appl..

[11]  Wim Michiels,et al.  An inner convex approximation algorithm for BMI optimization and applications in control , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[12]  Masakazu Kojima,et al.  Semidefinite Programming Relaxation for Nonconvex Quadratic Programs , 1997, J. Glob. Optim..

[13]  James E. Falk,et al.  Jointly Constrained Biconvex Programming , 1983, Math. Oper. Res..

[14]  S. Low,et al.  Zero Duality Gap in Optimal Power Flow Problem , 2012, IEEE Transactions on Power Systems.

[15]  Alan J. Miller.  Subset Selection in Regression , 1992 .

[16]  Emmanuel J. Candès,et al.  Exact Matrix Completion via Convex Optimization , 2009, Found. Comput. Math..

[17]  John E. Mitchell,et al.  Convex quadratic relaxations of nonconvex quadratically constrained quadratic programs , 2014, Optim. Methods Softw..

[18]  Amir Ali Ahmadi,et al.  DSOS and SDSOS optimization: LP and SOCP-based alternatives to sum of squares optimization , 2014, 2014 48th Annual Conference on Information Sciences and Systems (CISS).

[19]  Alberto Suárez,et al.  Optimization Problems with Cardinality Constraints , 2010 .

[20]  J. Lasserre,et al.  Handbook on Semidefinite, Conic and Polynomial Optimization , 2012 .

[21]  Laurent El Ghaoui,et al.  Rank Minimization under LMI constraints: A Framework for Output Feedback Problems , 2007 .

[22]  Pablo A. Parrilo,et al.  Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization , 2007, SIAM Rev..

[23]  Stephen P. Boyd,et al.  Enhancing Sparsity by Reweighted ℓ1 Minimization , 2007, 0711.1612.

[24]  Lorenz T. Biegler,et al.  On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming , 2006, Math. Program..

[25]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[26]  Stephen P. Boyd,et al.  Disciplined Convex Programming , 2006 .

[27]  K. Fan.  On a Theorem of Weyl Concerning Eigenvalues of Linear Transformations I. , 1949, Proceedings of the National Academy of Sciences of the United States of America.

[28]  K. Fan.  On a Theorem of Weyl Concerning Eigenvalues of Linear Transformations: II. , 1949, Proceedings of the National Academy of Sciences of the United States of America.