Chee-Khian Sim

Underlying paths in interior point methods for the monotone semidefinite linear complementarity problem

An interior point method defines a search direction at each interior point of the feasible region. The search directions at all interior points together form a direction field, which gives rise to a system of ordinary differential equations (ODEs). Given an initial point in the interior of the feasible region, the unique solution of the ODE system is a curve passing through the point, with tangents parallel to the search directions along the curve. We call such curves off-central paths. We study off-central paths for the monotone semidefinite linear complementarity problem (SDLCP). We show that each off-central path is a well-defined analytic curve with parameterxa0μ ranging over (0, ∞) and any accumulation point of the off-central path is a solution to SDLCP. Through a simple example we show that the off-central paths are not analytic as a function of

A Subgradient Method Based on Gradient Sampling for Solving Convex Optimization Problems

sqrt{mu}

Inexact subgradient methods for quasi-convex optimization problems

and have first derivatives which are unbounded as a function ofxa0μ atxa0μ xa0=xa0 0 in general. On the other hand, for the same example, we can find a subset of off-central paths which are analytic atxa0μ xa0=xa0 0. These “nice” paths are characterized by some algebraic equations.

The impact of supply chain visibility when lead time is random

Based on a formula of Tseng, we show that the squared norm of the matrix-valued Fischer-Burmeister function has a Lipschitz continuous gradient.

Optimal policies for inventory systems with two types of product sharing common hardware platforms:single period and finite horizon

Based on the gradient sampling technique, we present a subgradient algorithm to solve the nondifferentiable convex optimization problem with an extended real-valued objective function. A feature of our algorithm is the approximation of subgradient at a point via random sampling of (relative) gradients at nearby points, and then taking convex combinations of these (relative) gradients. We prove that our algorithm converges to an optimal solution with probability 1. Numerical results demonstrate that our algorithm performs favorably compared with existing subgradient algorithms on applications considered.

Profit sharing agreements in decentralized supply chains: a distributionally robust approach

Abstract.It is well known that a vector is in a second order cone if and only if its “arrow” matrix is positive semidefinite. But much less well-known is about the relation between a second order cone program (SOCP) and its corresponding semidefinite program (SDP). The correspondence between the dual problem of SOCP and SDP is quite direct and the correspondence between the primal problems is much more complicated. Given a SDP primal optimal solution which is not necessarily “arrow-shaped”, we can construct a SOCP primal optimal solution. The mapping from the primal optimal solution of SDP to the primal optimal solution of SOCP can be shown to be unique. Conversely, given a SOCP primal optimal solution, we can construct a SDP primal optimal solution which is not an “arrow” matrix. Indeed, in general no primal optimal solutions of the SOCP-related SDP can be an “arrow” matrix.

Superlinear Convergence of an Infeasible Predictor-Corrector Path-Following Interior Point Algorithm for a Semidefinite Linear Complementarity Problem Using the Helmberg-Kojima-Monteiro Direction

In this paper, we consider a generic inexact subgradient algorithm to solve a nondifferentiable quasi-convex constrained optimization problem. The inexactness stems from computation errors and noise, which come from practical considerations and applications. Assuming that the computational errors and noise are deterministic and bounded, we study the effect of the inexactness on the subgradient method when the constraint set is compact or the objective function has a set of generalized weak sharp minima. In both cases, using the constant and diminishing stepsize rules, we describe convergence results in both objective values and iterates, and finite convergence to approximate optimality. We also investigate efficiency estimates of iterates and apply the inexact subgradient algorithm to solve the Cobb–Douglas production efficiency problem. The numerical results verify our theoretical analysis and show the high efficiency of our proposed algorithm, especially for the large-scale problems.

Infinite horizon optimal policy for an inventory system with two types of product sharing common hardware platforms

We study the impact of supply chain visibility on periodic review inventory control, by studying its effect on lead time, where lead time is assumed to be random. We break down the total lead time from a supplier to a retailer into individual smaller lead times, as the product is tracked moving from one intermediary location to another. (This can be achieved, for example, through the use of RFID technology.) Under optimality, we observe that the average expected cost per cycle that the retailer can achieve in the long run given supply chain visibility is no worse than that for the base periodic review model without such visibility. An example is given which shows that there is indeed cost savings in the former, as compared with the base model. Further numerical results are then given to quantify the benefits of supply chain visibility on retailer’s cost with defined simple lead-time distributions. We also report on any trends that might appear as input parameters are varied in the numerical experiments.