Alternating Proximal Point Algorithm with Gradient Descent and Ascent Steps for Convex-Concave Saddle-Point Problems

Inspired by the Optimistic Gradient Ascent-Proximal Point Algorithm (OGAProx) proposed by Bo{\c{t}}, Csetnek, and Sedlmayer for solving a saddle-point problem associated with a convex-concave function with a nonsmooth coupling function and one regularizing function, we introduce the Alternating Proximal Point Algorithm with Gradient Descent and Ascent Steps for solving a saddle-point problem associated with a convex-concave function constructed by a smooth coupling function and two regularizing functions. In this work, we not only provide weak and linearly convergence of the sequence of iterations and of the minimax gap function evaluated at the ergodic sequences, similarly to what Bo{\c{t}} et al.\,did, but also demonstrate the convergence and linearly convergence of function values evaluated at convex combinations of iterations under convex and strongly convex assumptions, respectively.Comment: 25 page

Alternating Proximity Mapping Method for Strongly Convex-Strongly Concave Saddle-Point Problems

This is a continuation of our previous work entitled \enquote{Alternating Proximity Mapping Method for Convex-Concave Saddle-Point Problems}, in which we proposed the alternating proximal mapping method and showed convergence results on the sequence of our iterates, the sequence of averages of our iterates, and the sequence of function values evaluated at the averages of the iterates for solving convex-concave saddle-point problems. In this work, we extend the application of the alternating proximal mapping method to solve strongly convex-strongly concave saddle-point problems. We demonstrate two sets of sufficient conditions and also their simplified versions, which guarantee the linear convergence of the sequence of iterates towards a desired saddle-point. Additionally, we provide two sets of sufficient conditions, along with their simplified versions, that ensure the linear convergence of the sequence of function values evaluated at the convex combinations of iteration points to the desired function value of a saddle-point.Comment: 32 page

Paradise In The East

“Paradise in the East” is a 3D animation that explores the use of 3D character design, rigging, animation and landscaping to create a hyper-realistic look of some of China’s astonishingly diverse landscapes. This project referenced some of China’s most recognizable locations such as The Great Wall, Zhang Jiajie National Park, West Lake, etc

Alternating Proximity Mapping Method for Convex-Concave Saddle-Point Problems

We proposed an iterate scheme for solving convex-concave saddle-point problems associated with general convex-concave functions. We demonstrated that when our iterate scheme is applied to a special class of convex-concave functions, which are constructed by a bilinear coupling term plus a difference of two convex functions, it becomes a generalization of several popular primal-dual algorithms from constant involved parameters to involved parameters as general sequences. For this specific class of convex-concave functions, we proved that the sequence of function values, taken over the averages of iterates generated by our scheme, converges to the value of the function at a saddle-point. Additionally, we provided convergence results for both the sequence of averages of our iterates and the sequence of our iterates. In our numerical experiments, we implemented our algorithm in a matrix game, a linear program in inequality form, and a least-squares problem with $\ell_{1}$ regularization. In these examples, we also compared our algorithm with other primal-dual algorithms where parameters in their iterate schemes were kept constant. Our experimental results validated our theoretical findings. Furthermore, based on our experiments, we observed that when we consider one of the three examples mentioned above, it is either the sequence of function evaluations at the averages of iterates or the sequence of function evaluations at the iterates outperforms the other, regardless of changes in iterate schemes, problem data, and initial points. We also noted that certain involved parameters do not affect convergence rates in some instances, and that our algorithm consistently performs very well when compared to various iterate schemes with constant involved parameters.Comment: 35 pages and 11 figure