arXiv subjects

Chee-Khian Sim

Publications and source records attributed to Chee-Khian Sim.

7 recordsLinked to original sources

arXiv2025

First Order Algorithm on an Optimization Problem with Improved Convergence when Problem is Convex

We propose a first order algorithm, a modified version of FISTA, to solve an optimization problem with an objective function that is a sum of a possibly nonconvex function, with Lipschitz continuous gradient, and a convex function which can be nonsmooth. The algorithm is shown to have an iteration complexity of $\mathcal{O}(\epsilon^{-2})$ to find an $\epsilon$-approximate solution to the problem, and this complexity improves to $\mathcal{O}(\epsilon^{-2/3})$ when the objective function turns out to be convex. We further provide asymptotic convergence rate for the algorithm of worst case $o(\epsilon^{-2})$ iterations to find an $\epsilon$-approximate solution to the problem, with worst case $o(\epsilon^{-2/3})$ iterations when its objective function is convex.

math.OC↗

arXiv2024

Reduction from the partition problem: Dynamic lot sizing problem with polynomial complexity

In this note, we polynomially reduce an instance of the partition problem to a dynamic lot sizing problem, and show that solving the latter problem solves the former problem. By solving the dynamic program formulation of the dynamic lot sizing problem, we show that the instance of the partition problem can be solved with pseudo-polynomial time complexity. Numerical results on solving instances of the partition problem are also provided using an implementation of the algorithm that solves the dynamic program. We conclude by discussing polynomial time solvability of the partition problem through further observation on the dynamic program formulation of the dynamic lot sizing problem.

cs.CC↗

arXiv2024

Refining asymptotic complexity bounds for nonconvex optimization methods, including why steepest descent is $o(\epsilon^{-2})$ rather than $\mathcal{O}(\epsilon^{-2})$

We revisit the standard ``telescoping sum'' argument ubiquitous in the final steps of analyzing evaluation complexity of algorithms for smooth nonconvex optimization, and obtain a refined formulation of the resulting bound as a function of the requested accuracy $\epsilon$. While bounds obtained using the standard argument typically are of the form $\mathcal{O}(\epsilon^{-\alpha})$ for some positive $\alpha$, the refined results are of the form $o(\epsilon^{-\alpha})$. We then explore to which known algorithms our refined bounds are applicable and finally describe an example showing how close the standard and refined bounds can be.

math.OC↗

arXiv2022

Superlinear Convergence of an Interior Point Algorithm on Linear Semi-definite Feasibility Problems

In the literature, besides the assumption of strict complementarity, superlinear convergence of implementable polynomial-time interior point algorithms using known search directions, namely, the HKM direction, its dual or the NT direction, to solve semi-definite programs (SDPs) is shown by (i) assuming that the given SDP is nondegenerate and making modifications to these algorithms [10], or (ii) considering special classes of SDPs, such as the class of linear semi-definite feasibility problems (LSDFPs) and requiring the initial iterate to the algorithm to satisfy certain conditions [26, 27]. Otherwise, these algorithms are not easy to implement even though they are shown to have polynomial iteration complexities and superlinear convergence [14]. The conditions in [26, 27] that the initial iterate to the algorithm is required to satisfy to have superlinear convergence when solving LSDFPs however are not practical. In this paper, we propose a practical initial iterate to an implementable infeasible interior point algorithm that guarantees superlinear convergence when the algorithm is used to solve the homogeneous feasibility model of an LSDFP.

math.OC↗

arXiv2020

A FISTA-Type First Order Algorithm on Composite Optimization Problems that is Adaptable to the Convex Situation

In this note, we propose a FISTA-type first order algorithm, VAR-FISTA, to solve a composite optimization problem. A distinctive feature of VAR-FISTA is its ability to exploit the convexity of the function in the problem, resulting in an improved iteration complexity when the function is convex compared to when it is nonconvex. The iteration complexity result for the convex and nonconvex case obtained in the note are compatible to the best known in the literature so far.

math.OC↗

arXiv2019

A FISTA-type accelerated gradient algorithm for solving smooth nonconvex composite optimization problems

In this paper, we describe and establish iteration-complexity of two accelerated composite gradient (ACG) variants to solve a smooth nonconvex composite optimization problem whose objective function is the sum of a nonconvex differentiable function $ f $ with a Lipschitz continuous gradient and a simple nonsmooth closed convex function $ h $. When $f$ is convex, the first ACG variant reduces to the well-known FISTA for a specific choice of the input, and hence the first one can be viewed as a natural extension of the latter one to the nonconvex setting. The first variant requires an input pair $(M,m)$ such that $f$ is $m$-weakly convex, $\nabla f$ is $M$-Lipschitz continuous, and $m \le M$ (possibly $m<M$), which is usually hard to obtain or poorly estimated. The second variant on the other hand can start from an arbitrary input pair $(M,m)$ of positive scalars and its complexity is shown to be not worse, and better in some cases, than that of the first variant for a large range of the input pairs. Finally, numerical results are provided to illustrate the efficiency of the two ACG variants.

math.OC↗

arXiv2016

Complexity of the relaxed Peaceman-Rachford splitting method for the sum of two maximal strongly monotone operators

This paper considers the relaxed Peaceman-Rachford (PR) splitting method for finding an approximate solution of a monotone inclusion whose underlying operator consists of the sum of two maximal strongly monotone operators. Using general results obtained in the setting of a non-Euclidean hybrid proximal extragradient framework, we extend a previous convergence result on the iterates generated by the relaxed PR splitting method, as well as establish new pointwise and ergodic convergence rate results for the method whenever an associated relaxation parameter is within a certain interval. An example is also discussed to demonstrate that the iterates may not converge when the relaxation parameter is outside this interval.

math.OC↗