Computation of the Epsilon-Subdifferential of Convex Piecewise Linear-Quadratic Functions in Optimal Worst-Case Time

Set-Valued and Variational Analysis - The epsilon-subdifferential of convex univariate piecewise linear-quadratic (PLQ) functions can be computed in linear worst-case time complexity as the...     

Computing the conjugate of convex piecewise linear-quadratic bivariate functions

We present a new algorithm to compute the Legendre–Fenchel conjugate of convex piecewise linear-quadratic (PLQ) bivariate functions. The algorithm stores a function using a (primal) planar arrangement. It then computes the (dual) arrangement associated with the conjugate by looping through vertices, edges, and faces in the primal arrangement and building associated dual vertices, edges, and faces. Using optimal computational geometry data structures, the algorithm has a linear time worst-case complexity. We present the algorithm, and illustrate it with numerical examples. We proceed to build a toolbox for convex bivariate PLQ functions by implementing the addition, and scalar multiplication operations. Finally, we compose these operators to compute classical convex analysis operators such as the Moreau envelope, and the proximal average.     

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

Computing the convex envelope is a core operation in nonsmooth analysis that bridges the convex with the nonconvex world. Although efficient algorithms to compute fundamental transforms of convex analysis have been proposed over the years, they are limited to convex functions until an efficient algorithm becomes available to compute the convex envelope of a piecewise linear-quadratic function (of one variable) efficiently. We present two such 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 implement but has optimal (linear time) complexity. We prove their time (and space) complexity, and compare their performances.     

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. This revised version was published online in June 2006 with corrections to the Cover Date.     

Hybrid symbolic-numeric algorithms for computational convex analysis

Computational convex analysis focuses on developing efficient tools to compute fundamental transforms arising in convex analysis. Symbolic computation tools have been developed, and have allowed more insight into the calculation of the Fenchel conjugate and related transforms. When such tools are not applicable e.g. when there is no closed form, fast transform algorithms perform numerical computation efficiently. However, computing the composition of several transforms is difficult to achieve with fast transform algorithms, which is the case for the recently introduced proximal average operator. We consider the class of piecewise linear-quadratic functions which, being closed under the most relevant operations in convex analysis, allows the robust and efficient numerical computation of compositions of transforms like the proximal average. The algorithms presented are hybrid symbolic-numeric: they first compute a piecewise linear-quadratic approximation of the function, and then manipulate the approximation symbolically. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A linear-time algorithm to compute the conjugate of convex piecewise linear-quadratic bivariate functions

We propose the first algorithm to compute the conjugate of a bivariate Piecewise Linear-Quadratic (PLQ) function in optimal linear worst-case time complexity. The key step is to use a planar graph, called the entity graph, not only to represent the entities (vertex, edge, or face) of the domain of a PLQ function but most importantly to record adjacent entities. We traverse the graph using breadth-first search to compute the conjugate of each entity using graph-matrix calculus, and use the adjacency information to create the output data structure in linear time.     

Fast Moreau envelope computation I: numerical algorithms

The present article summarizes the state of the art algorithms to compute the discrete Moreau envelope, and presents a new linear-time algorithm, named NEP for NonExpansive Proximal mapping. Numerical comparisons between the NEP and two existing algorithms: The Linear-time Legendre Transform (LLT) and the Parabolic Envelope (PE) algorithms are performed. Worst-case time complexity, convergence results, and examples are included. The fast Moreau envelope algorithms first factor the Moreau envelope as several one-dimensional transforms and then reduce the brute force quadratic worst-case time complexity to linear time by using either the equivalence with Fast Legendre Transform algorithms, the computation of a lower envelope of parabolas, or, in the convex case, the non expansiveness of the proximal mapping.      

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

Piecewise linear-quadratic (PLQ) functions are an important class of functions in convex analysis since the result of most convex operators applied to a PLQ function is a PLQ function. We modify a recent algorithm for computing the convex (Legendre-Fenchel) conjugate of convex PLQ functions of two variables, to compute its partial conjugate i.e. the conjugate with respect to one of the variables. The structure of the original algorithm is preserved including its time complexity (linear time with some approximation and log-linear time without approximation). Applying twice the partial conjugate (and a variable switching operator) recovers the full conjugate. We present our partial conjugate algorithm, which is more flexible and simpler than the original full conjugate algorithm. We emphasize the difference with the full conjugate algorithm and illustrate results by computing partial conjugates, partial Moreau envelopes, and partial proximal averages.     

The piecewise linear-quadratic model for computational convex analysis

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.     

Convexity of the Proximal Average

We complete the study of the convexity of the proximal average by proving it is convex as a function of each of its parameters separately, but not jointly convex as a function of any two of its parameters. We present an interpolation-based plotting algorithm that takes advantage of the partial convexity of the proximal average, and improves the plotting time by a factor of 100, while reducing picture sizes by a factor of 10.