Yves Lucet

The Proximal Average: Basic Theory

The recently introduced proximal average of two convex functions is a convex function with many useful properties. In this paper, we introduce and systematically study the proximal average for finitely many convex functions. The basic properties of the proximal average with respect to the standard convex-analytical notions (domain, Fenchel conjugate, subdifferential, proximal mapping, epi-continuity, and others) are provided and illustrated by several examples.

Faster than the Fast Legendre Transform, the Linear-time Legendre Transform

A new algorithm to compute the Legendre–Fenchel transform is proposed and investigated. The so-called Linear-time Legendre Transform (LLT) improves all previously known Fast Legendre Transform algorithms by reducing their log-linear worst-case time complexity to linear. Since the algorithm amounts to computing several convex hulls and sorting, any convex hull algorithm well-suited for a particular problem gives a corresponding LLT algorithm. After justifying the convergence of the Discrete Legendre Transform to the Legendre–Fenchel transform, an extended computation time complexity analysis is given and confirmed by numerical tests. Finally, the LLT is illustrated with several examples and a LLT MATLAB package is described.

A fast computational algorithm for the Legendre-Fenchel transform

We investigate a fast algorithm, introduced by Brenier, which computes the Legendre-Fenchel transform of a real-valued function. We generalize his work to boxed domains and introduce a parameter in order to build an iterative algorithm. The new approach of separating primal and dual spaces allows a clearer understanding of the algorithm and yields better numerical behavior. We extend known complexity results and give new ones about the convergence of the algorithm.

What Shape Is Your Conjugate? A Survey of Computational Convex Analysis and Its Applications

Computational convex analysis algorithms have been rediscovered several times in the past by researchers from different fields. To further communications between practitioners, we review the field of computational convex analysis, which focuses on the numerical computation of fundamental transforms arising from convex analysis. Current models use symbolic, numeric, and hybrid symbolic-numeric algorithms. Our objective is to disseminate widely the most efficient numerical algorithms useful for applications in image processing (computing the distance transform, the generalized distance transform, and mathematical morphology operators), partial differential equations (solving Hamilton-Jacobi equations and using differential equations numerical schemes to compute the convex envelope), max-plus algebra (computing the equivalent of the fast Fourier transform), multifractal analysis, etc. The fields of applications include, among others, computer vision, robot navigation, thermodynamics, electrical networks, medical imaging, and network communication.

How to Transform One Convex Function Continuously into Another

The proximal average operator provides a parametric family of convex functions that continuously transform one convex function into another even when the domains of the two functions do not intersect. We prove that the proximal average operator is a homotopy with respect to the epi-topology, study its properties, and present several explicit formulas for specific classes of functions. The parametric family inherits desirable properties such as differentiability and strict convexity from the given functions. The results illustrate the powerful tools available in convex and variational analysis from both a theoretical and a computational point of view.

Sensitivity Analysis of the Value Function for Optimization Problems with Variational Inequality Constraints

In this paper we perform sensitivity analysis for optimization problems with variational inequality constraints (OPVICs). We provide upper estimates for the limiting subdifferential (singular limiting subdifferential) of the value function in terms of the set of normal (abnormal) coderivative (CD) multipliers for OPVICs. For the case of optimization problems with complementarity constraints (OPCCs), we provide upper estimates for the limiting subdifferentials in terms of various multipliers. An example shows that the other multipliers may not provide useful information on the subdifferentials of the value function, while the CD multipliers may provide tighter bounds. Applications to sensitivity analysis of bilevel programming problems are also given.

The piecewise linear-quadratic model for computational convex analysis

Abstract A new computational framework for computer-aided convex analysis is proposed and investigated. Existing computational frameworks are reviewed and their limitations pointed out. The class of piecewise linear-quadratic functions is introduced to improve convergence and stability. A stable convex calculus is achieved using symbolic-numeric algorithms to compute all fundamental transforms of convex analysis. Our main result states the existence of efficient (linear time) algorithms for the class of piecewise linear-quadratic functions. We also recall that such class is closed under convex transforms. We illustrate the results with numerical examples, and validate numerically the resulting computational framework.

Primal-Dual Symmetric Intrinsic Methods for Finding Antiderivatives of Cyclically Monotone Operators

A fundamental result due to Rockafellar states that every cyclically monotone operator

Fast Moreau envelope computation I: numerical algorithms

A

New sequential exact Euclidean distance transform algorithms based on convex analysis

admits an antiderivative