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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.

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) [Raza SA (2014) A distribution free approach to newsvendor problem with pricing. 4OR—A Quart. J. Oper. Res. 12(4):335–358.]. The online appendix is available at https://doi.org/10.1287/opre.2017.1677 .

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 parameter ... ranging over ... 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 ... and have first derivatives which are unbounded as a function of ... at ... in general. On the other hand, for the same example, we can find a subset of off-central paths which are analytic at ... . These \"nice\"\" paths are characterized by some algebraic equations. (ProQuest: ... denotes formulae omitted)

An interior point method (IPM) defines a search direction at each interior point of a region. These search directions form a direction field which in turn gives rise to a system of ordinary differential equations (ODEs). The solutions of the system of ODEs can be viewed as underlying paths in the interior of the region. In [C.-K. Sim and G. Zhao, Math. Program. Ser. A, 110 (2007), pp. 475-499], these off-central paths are shown to be well-defined analytic curves, and any of their accumulation points is a solution to a given monotone semidefinite linear complementarity problem (SDLCP). The study of these paths provides a way to understand how iterates generated by an interior point algorithm behave. In this paper, we give a sufficient condition using these off-central paths that guarantees superlinear convergence of a predictor-corrector path-following interior point algorithm for SDLCP using the Helmberg-Kojima-Monteiro (HKM) direction. This sufficient condition is implied by a currently known sufficient condition for superlinear convergence. Using this sufficient condition, we show that for any linear semidefinite feasibility problem, superlinear convergence using the interior point algorithm, with the HKM direction, can be achieved for a suitable starting point. We work under the assumption of strict complementarity. [PUBLICATION ABSTRACT]

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

An interior point method (IPM) defines a search direction at each interior point of the feasible region. These search directions form a direction field, which in turn gives rise to a system of ordinary differential equations (ODEs). Thus, it is natural to define the underlying paths of the IPM as solutions of the system of ODEs. In Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007 ), these off-central paths are shown to be well-defined analytic curves and any of their accumulation points is a solution to the given monotone semidefinite linear complementarity problem (SDLCP). In Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007 ; J. Optim. Theory Appl. 137:11–25, 2008 ) and Sim (J. Optim. Theory Appl. 141:193–215, 2009 ), the asymptotic behavior of off-central paths corresponding to the HKM direction is studied. In particular, in Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007 ), the authors study the asymptotic behavior of these paths for a simple example, while, in Sim and Zhao (J. Optim. Theory Appl. 137:11–25, 2008 ) and Sim (J. Optim. Theory Appl. 141:193–215, 2009 ), the asymptotic behavior of these paths for a general SDLCP is studied. In this paper, we study off-central paths corresponding to another well-known direction, the Nesterov-Todd (NT) direction. Again, we give necessary and sufficient conditions for these off-central paths to be analytic w.r.t. and then w.r.t. μ , at solutions of a general SDLCP. Also, as in Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007 ), we present off-central path examples using the same SDP, whose first derivatives are likely to be unbounded as they approach the solution of the SDP. We work under the assumption that the given SDLCP satisfies a strict complementarity condition.

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.