2008/04/10 by Néstor E. Aguilera, Aguilera, Néstor, Pedro Morín +1
Computer Science · Decision Sciences · Mathematics · #65K10 #65N30 #Advanced Optimization Algorithms Research #FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Variational Analysis #Probabilistic and Robust Engineering Design
paper · pdf · doi:10.48550/arxiv.0804.1780
openalex publication_date 2008/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Many problems of theoretical and practical interest involve finding a convex or concave function. For instance, optimization problems such as finding the projection on the convex functions in Hk(Ω), or some problems in economics. In the continuous setting and assuming smoothness, the convexity constraints may be given locally by asking the Hessian matrix to be positive semidefinite, but in making discrete approximations two difficulties arise: the continuous solutions may be not smooth, and an adequate discrete version of the Hessian must be given. In this paper we propose a finite element description of the Hessian, and prove convergence under very general conditions, even when the continuous solution is not smooth, working on any dimension, and requiring a linear number of constraints in the number of nodes. Using semidefinite programming codes, we show concrete examples of approximations to optimization problems.