necessary and sufficient conditions

Sufficient Conditions for Low-rank Matrix Recovery,Translated from Sparse Signal Recovery

Published: 2011/06/15

Convex and Nonsmooth Optimization, Nonlinear Optimization hBcgoodness, low-rank matrix recovery, necessary and sufficient conditions, restricted isometry constant

The low-rank matrix recovery (LMR) is a rank minimization problem subject to linear equality constraints, and it arises in many fields such as signal and image processing, statistics, computer vision, system identification and control. This class of optimization problems is $NP$-hard and a popular approach replaces the rank function with the nuclear norm of the … Read more

Transposition theorems and qualification-free optimality conditions

Published: 2005/05/11, Updated: 2006/02/20

Arnold Neumaier

Hermann Schichl

Constrained Nonlinear Optimization, Global Optimization Theory, Multi-Criteria Optimization certificate of global optimality, first order optimality conditions, fritz john conditions, global optimality condition, global optimization, karush-john conditions, kuhn-tucker conditions, mangasarian-fromovitz constraint qualification, necessary and sufficient conditions, positivstellensatz, second order optimality conditions, theorem of the alternative, transposition theorem

New theorems of the alternative for polynomial constraints (based on the Positivstellensatz from real algebraic geometry) and for linear constraints (generalizing the transposition theorems of Motzkin and Tucker) are proved. Based on these, two Karush-John optimality conditions — holding without any constraint qualification — are proved for single- or multi-objective constrained optimization problems. The first … Read more