Chee-Khian Sim In this note, we reduce an instance of the partition problem to a dynamic lot sizing problem in polynomial time, and show that solving the latter problem solves the former problem. We further show that the instance of the partition problem can be solved using polynomial number of addition, multiplication and sort operations in input … Read more
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 eps. While bounds obtained using the standard argument typically are of the form \(O(\epsilon^{-\alpha})\) for some positive … Read more
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 … Read more
We consider the convergence behavior using the relaxed Peaceman-Rachford splitting method to solve the monotone inclusion problem $0 \in (A + B)(u)$, where $A, B: \Re^n \rightrightarrows \Re^n$ are maximal $\beta$-strongly monotone operators, $n \geq 1$ and $\beta > 0$. Under a technical assumption, convergence of iterates using the method on the problem is proved … Read more
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 … Read more
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 … Read more
We consider in this paper an infeasible predictor-corrector primal-dual path following interior point algorithm using the Nesterov-Todd (NT) search direction to solve semi-definite linear complementarity problems. Global convergence and polynomial iteration complexity of the algorithm are established. Two sufficient conditions are also given for superlinear convergence of iterates generated by the algorithm. Preliminary numerical results … Read more
Finite horizon periodic review backlog models are considered in this paper for an inventory system that remanufactures two types of cores: buyback cores and normal cores. Returns of used products as buyback cores are modelled to depend on past demands and past sales. We obtain an optimal inventory policy for the model in which returns … Read more
This paper considers the relaxed Peaceman-Rachford (PR) splitting method for fi nding 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, convergence of the iterates, as well as pointwise and ergodic … Read more
We consider a periodic review inventory system and present its optimal policy in the infinite horizon setting. The optimal inventory policy that maximizes the infinite horizon expected discount profit for the model is analytically obtained by relating to the finite horizon setting using results from variational analysis. Results are provided that elucidate the operations of … Read more