Projections onto the canonical simplex with additional linear inequalities. (4th March 2022)

• Record Type: Journal Article
• Authors: Adam, L.; Mácha, V.
• Is Part Of: Optimization methods and software. Volume 37:Number 2(2022)
• Page Start: 451
• Page End: 479
• Publication Date: 2022-03-04
• Subjects: Projection -- simplex -- distributionally robust optimization -- linear observed complexity; 90C20 -- 49M05 -- 65K10 -- 49K10; Mathematical optimization -- Periodicals; Algorithms -- Periodicals; 519.7
• DOI: 10.1080/10556788.2020.1797023
• Languages: English
• ISSNs: 1055-6788

Abstract

Abstract : We consider the distributionally robust optimization and show that computing the distributional worst-case is equivalent to computing the projection onto the canonical simplex with additional linear inequality. We consider several distance functions to measure the distance of distributions. We write the projections as optimization problems and show that they are equivalent to finding a zero of real-valued functions. We prove that these functions possess nice properties such as monotonicity or convexity. We design optimization methods with guaranteed convergence and derive their theoretical complexity. We demonstrate that our methods have (almost) linear observed complexity.