1,397 research outputs found
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
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
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