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Applications of gauge duality in robust principal component analysis and semidefinite programming. (English) Zbl 1380.65106
The nonlinear gauge optimization problem is to minimize a closed gauge function $$\kappa$$ over a closed convex set $$X \subset \mathbb{R}^n$$, i.e. $$\min_x\{\kappa(x)|x\in X\}$$. Its nonlinear gauge dual problem is defined as minimizing the polar function $$\kappa^\circ$$ over the antipolar set $$X'$$, i.e. $$\min_y\{\kappa^\circ(y)|y\in X'\}$$, where $$\kappa^\circ(y)=\inf\{\mu>0|\langle x,y\rangle \leq \mu\kappa(x) \text{ for all } x\}$$. The authors present new theoretical results on applying the gauge duality theory [R. M. Freund, Math. Program. 38, 47–67 (1987; Zbl 0632.90054)] to robust principal component analusis and general semidefinite programming [M. P. Friedlander and I. Macêdo, SIAM J. Sci. Comput. 38, No. 3, A1616–A1638 (2016; Zbl 1342.90115)]. For each considered problem, they give its gauge dual problem, characterize the optimality conditions for the primal-dual gauge pair and finally they validate a way to recover a primal optimal solution from a dual one.
##### MSC:
 65K05 Numerical mathematical programming methods 90C25 Convex programming 90C22 Semidefinite programming 60H25 Random operators and equations (aspects of stochastic analysis)