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An interior point method (IPM) 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 parameter µ 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 √ µ and have first derivatives which are unbounded as a function of µ at µ = 0 in general. On the other hand, for the same example, we can find a subset of off-central paths which are analytic at µ = 0. These "nice" paths are characterized by some algebraic equations.

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 as input a pair (M, m), M being a Lipschitz constant of ∇f and m being a lower curvature of f such that m ≤ 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.

How should decentralized supply chains set the profit sharing terms using minimal information on demand and selling price? We develop a distributionally robust Stackelberg game model to address this question. Our framework uses only the first and second moments of the price and demand attributes, and thus can be implemented using only a parsimonious set of parameters. More specifically, we derive the relationships among the optimal wholesale price set by the supplier, the order decision of the retailer, and the corresponding profit shares of each supply chain partner, based on the information available. Interestingly, in the distributionally robust setting, the correlation between demand and selling price has no bearing on the order decision of the retailer. This allows us to simplify the solution structure of the profit sharing agreement problem dramatically. Moreover, the result can be used to recover the optimal selling price when the mean demand is a linear function of the selling price (cf. Raza (2014)).

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