Robust linear optimization under matrix completion

Linear programming models have been widely used in input-output analysis for analyzing the interdependence of industries in economics and in environmental science. In these applications, some of the entries of the coefficient matrix cannot be measured physically or there exists sampling errors. However, the coefficient matrix can often be low-rank. We characterize the robust counterpart of these types of linear programming problems with uncertainty set described by the nuclear norm. Simulations for the input-output analysis show that the new paradigm can be helpful.

[1]  Geoffrey J. D. Hewings,et al.  Structural change decomposition through a global sensitivity analysis of input–output models , 2006 .

[2]  Chen Lin,et al.  Identifying Lowest‐Emission Choices and Environmental Pareto Frontiers for Wastewater Treatment Wastewater Treatment Input‐Output Model based Linear Programming , 2011 .

[3]  Victor Vianu,et al.  Invited articles section foreword , 2010, JACM.

[4]  Peter D. Blair,et al.  Input-Output Analysis , 2021 .

[5]  Laurent El Ghaoui,et al.  Robust Solutions to Least-Squares Problems with Uncertain Data , 1997, SIAM J. Matrix Anal. Appl..

[6]  R. Glowinski,et al.  Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics , 1987 .

[7]  B. He,et al.  Alternating Direction Method with Self-Adaptive Penalty Parameters for Monotone Variational Inequalities , 2000 .

[8]  Constantine Caramanis,et al.  Theory and Applications of Robust Optimization , 2010, SIAM Rev..

[9]  Adisa Azapagic,et al.  Life cycle assessment and linear programming environmental optimisation of product system , 1995 .

[10]  Arkadi Nemirovski,et al.  Robust optimization – methodology and applications , 2002, Math. Program..

[11]  Laurent El Ghaoui,et al.  Robust Solutions to Uncertain Semidefinite Programs , 1998, SIAM J. Optim..

[12]  Shinichiro Nakamura,et al.  Waste input–output linear programming model with its application to eco-efficiency analysis , 2005 .

[13]  Adisa Azapagic,et al.  Linear programming as a tool in life cycle assessment , 1998 .

[14]  W. Leontief.  Quantitative Input and Output Relations in the Economic Systems of the United States , 1936 .

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

[16]  Adisa Azapagic,et al.  Life cycle assessment and multiobjective optimisation , 1999 .

[17]  Emmanuel J. Candès,et al.  The Power of Convex Relaxation: Near-Optimal Matrix Completion , 2009, IEEE Transactions on Information Theory.

[18]  R. Bellman,et al.  Linear Programming and Economic Analysis. , 1960 .

[19]  Arkadi Nemirovski,et al.  Robust Convex Optimization , 1998, Math. Oper. Res..

[20]  Patrick L. Combettes,et al.  A Monotone+Skew Splitting Model for Composite Monotone Inclusions in Duality , 2010, SIAM J. Optim..

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

[22]  M. Fortin,et al.  Augmented Lagrangian methods : applications to the numerical solution of boundary-value problems , 1983 .

[23]  Yi Ma,et al.  Robust principal component analysis? , 2009, JACM.

[24]  Shiqian Ma,et al.  Fixed point and Bregman iterative methods for matrix rank minimization , 2009, Math. Program..

[25]  Wassily Leontief.  Input-Output Economics , 1966 .

[26]  Arkadi Nemirovski,et al.  Robust solutions of uncertain linear programs , 1999, Oper. Res. Lett..