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

@article{Lucet2009WhatSI, title={What Shape Is Your Conjugate? A Survey of Computational Convex Analysis and Its Applications}, author={Yves Lucet}, journal={SIAM Rev.}, year={2009}, volume={52}, pages={505-542} }

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…

## 74 Citations

### Computing the conjugate of convex piecewise linear-quadratic bivariate functions

- Mathematics, Computer ScienceMath. Program.
- 2013

A new algorithm to compute the Legendre–Fenchel conjugate of convex piecewise linear-quadratic (PLQ) bivariate functions and compose these operators to compute classical convex analysis operators such as the Moreau envelope, and the proximal average.

### Graph-Matrix Calculus for Computational Convex Analysis

- Computer Science, MathematicsFixed-Point Algorithms for Inverse Problems in Science and Engineering
- 2011

We introduce a new family of algorithms for computing fundamental operators arising from convex analysis. The new algorithms rely on the fact that the graph of the subdifferential of most convex…

### Computing the partial conjugate of convex piecewise linear-quadratic bivariate functions

- Mathematics, Computer ScienceComput. Optim. Appl.
- 2014

This work modifications a recent algorithm for computing the convex (Legendre-Fenchel) conjugate of convex PLQ functions of two variables, to compute its partial conjugates, which is more flexible and simpler than the original full conjugating algorithm.

### Convex Hull Algorithms for Piecewise Linear-Quadratic Functions in Computational Convex Analysis

- Computer Science, Mathematics
- 2010

This work presents two algorithms, one based on maximum and conjugate computation that is easy to implement but has quadratic time complexity, and another based on direct computation that requires more work to implementbut has optimal (linear time) complexity.

### Tropical Geometry and Piecewise-Linear Approximation of Curves and Surfaces on Weighted Lattices

- Mathematics, Computer ScienceArXiv
- 2019

This chapter provides the optimal solution of max-plus and min-plus equations using morphological adjunctions that are projections on weighted lattices, and applies it to optimal piecewise-linear regression for fitting max-$\star$ tropical curves and surfaces to arbitrary data that constitute polygonal or polyhedral shape approximations.

### Tropical Geometry and Machine Learning

- Computer Science, MathematicsProceedings of the IEEE
- 2021

This article summarizes introductory ideas and objects of tropical geometry, providing a theoretical framework for both the max-plus algebra that underlies tropical geometry and its extensions to general max algebras, and provides optimal solutions and an efficient algorithm for the convex regression problem.

### Computing the Subdifferential of Convex Piecewise Linear-Cubic Functions

- Computer Science
- 2021

A method is introduced that manipulates the PLCVC data structure in the CCA toolbox to plot an unbounded domain within a speciﬁed window and a class that implements a method computing the subdiﬀerential at any point of a PLC function is introduced.

### Techniques and Open Questions in Computational Convex Analysis

- Computer Science, Mathematics
- 2013

We present several techniques that take advantage of convexity and use optimal computational geometry algorithms to build fast (log-linear or linear) time algorithms in computational convex analysis.…

### Mathematical Morphology and Its Applications to Signal and Image Processing

- Computer Science, MathematicsLecture Notes in Computer Science
- 2019

This paper outlines the optimal solution of maxequations using weighted lattice adjunctions, and applies it to optimal regression for fitting maxtropical curves on arbitrary data.

### Hybrid symbolic‐numeric algorithms for computational convex analysis

- Mathematics, Computer Science
- 2007

Developing efficient tools to compute fundamental transforms arising in convex analysis and allowing more insight into the calculation of the Fenchel conjugate and related transforms are focused on.

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