This paper gives an approach to implementing a second-order primal-dual interior point method. It uses a Taylor polynomial of second order to approximate a primal-dual trajectory. The computations for the second derivative are combined with the computations for the centering direction. Computations in this approach do not require that primal and dual solutions be feasible. Expressions are given to compute all the higher-order derivatives of the trajectory of interest. The implementation ensures that a suitable potential function is reduced by a constant amount at each iteration.There are several salient features of this approach. An adaptive heuristic for estimating the centering parameter is given. The approach used to compute the step length is also adaptive. A new practical approach to compute the starting point is given. This approach treats primal and dual problems symmetrically.Computational results on a subset of problems available from netlib are given. On mutually tested problems the results show that the proposed method requires approximately 40 percent fewer iterations than the implementation proposed in Lustig, Marsten, and Shanno Tech. Rep. TR J-89-11, Georgia Inst. of Technology, Atlanta, 1989]. It requires approximately 50 percent fewer iterations than the dual affine scaling method in Adler, Karmarkar, Resende, and Veiga [Math. Programming, 44 (1989), pp. 297-336], and 35 percent fewer iterations than the second-order dual affine scaling method in the same paper. The new approach for estimating the centering parameter and finding the step length and the starting point have contributed to the reduction in the number of iterations. However, the contribution due to the use of second derivative is most significant.On the tested problems, on the average the implementation shown was found to be approximately two times faster than OB1 (version 02/90) described in Lustig, Marsten, and Shanno and 2.5 times faster than MINOS 5.3 described in Murtagh and Saunders A comparison with the results reported in the literature (on mutually tested problems) shows that the method developed in this paper takes approximately 50 percent fewer iterations than the dual affine scaling method as implemented by Adler, Karmarkar, Resende, and Veiga 1 ], 40 percent fewer iterations than the primal-dual method implemented in Lustig, Marsten, and Shanno [18], and 55 percent fewer iterations than the logarithmic barrier function method implemented in Gill, Murray, and Saunders [8]. It requires 35 percent fewer iterations than the second-order dual affine scaling method implemented in Adler, Karmarkar, Resende, and Veiga 1 and 20 percent fewer iterations than the "optimal three-dimensional method" implemented by Domich, Boggs, Donaldson, and Witzgall [5].An efficient preliminary implementation of the proposed approach was developed. On average, it was found to be two times faster than OB1 (version 02/1990) [18]. On average, it was also found to be 2.5 times faster than MINOS 5.3.

No abstract

No abstract

Going forward, the federal response to the COVID-19 pandemic will require centralized decision-making around how to equitably allocate, and reallocate, limited supplies of ventilators to states in need. Projections from the Institute for Health Metrics and Evaluation at the University of Washington, which assume that all states will institute strict social distancing practices and maintain them until after infections peak, show states will hit their peak demand at different time points throughout the months of April and May. Many states are predicted to experience a significant gap in ICU capacity, and similar, if not greater, gaps in ventilator capacity, with the time point at which needs will begin to exceed current capacity varying by state [8].

The CREATE-ECLA Randomized Controlled TrialThe CREATE-ECLA Trial Group Investigators* See also pp 427 and 489.

We study the problem of integrated staffing and scheduling under demand uncertainty. This problem is formulated as a two-stage stochastic integer program with mixed-integer recourse. The here-and-now decision is to find initial staffing levels and schedules. The wait-and-see decision is to adjust these schedules at a time closer to the actual date of demand realization. We show that the mixed-integer rounding inequalities for the second-stage problem convexify the recourse function. As a result, we present a tight formulation that describes the convex hull of feasible solutions in the second stage. We develop a modified multicut approach in an integer L-shaped algorithm with a prioritized branching strategy. We generate twenty instances (each with more than 1.3 million integer and 4 billion continuous variables) of the staffing and scheduling problem using 3.5 years of patient volume data from Northwestern Memorial Hospital. Computational results show that the efficiency gained from the convexification of the recourse function is further enhanced by our modifications to the L-shaped method. The results also show that compared with a deterministic model, the two-stage stochastic model leads to a significant cost savings. The cost savings increase with mean absolute percentage errors in the patient volume forecast.

We present and analyze a central cutting surface algorithm for general semi-infinite convex optimization problems, and use it to develop a novel algorithm for distributionally robust optimization problems in which the uncertainty set consists of probability distributions with given bounds on their moments. Moments of arbitrary order, as well as non-polynomial moments can be included in the formulation. We show that this gives rise to a hierarchy of optimization problems with decreasing levels of risk-aversion, with classic robust optimization at one end of the spectrum, and stochastic programming at the other. Although our primary motivation is to solve distributionally robust optimization problems with moment uncertainty, the cutting surface method for general semi-infinite convex programs is also of independent interest. The proposed method is applicable to problems with non-differentiable semi-infinite constraints indexed by an infinite-dimensional index set. Examples comparing the cutting surface algorithm to the central cutting plane algorithm of Kortanek and No demonstrate the potential of our algorithm even in the solution of traditional semi-infinite convex programming problems, whose constraints are differentiable, and are indexed by an index set of low dimension. After the rate of convergence analysis of the cutting surface algorithm, we extend the authors' moment matching scenario generation algorithm to a probabilistic algorithm that finds optimal probability distributions subject to moment constraints. The combination of this distribution optimization method and the central cutting surface algorithm yields a solution to a family of distributionally robust optimization problems that are considerably more general than the ones proposed to date.

In this paper, we introduce and study a two-stage distributionally robust mixed binary problem (TSDR-MBP) where the random parameters follow the worst-case distribution belonging to an uncertainty set of probability distributions. We present a decomposition algorithm, which utilizes distribution separation procedure and parametric cuts within Benders' algorithm or Lshaped method, to solve TSDR-MBPs with binary variables in the first stage and mixed binary programs in the second stage. We refer to this algorithm as distributionally robust integer (DRI) L-shaped algorithm. Using similar decomposition framework, we provide another algorithm to solve TSDR linear problem where both stages have only continuous variables. We investigate conditions and the families of ambiguity set for which our algorithms are finitely convergent. We present two examples of ambiguity set, defined using moment matching, or Kantorovich-Rubinstein distance (Wasserstein metric), which satisfy the foregoing conditions. We also present a cutting surface algorithm to solve TSDR-MBPs. We computationally evaluate the performance of the DRI Lshaped algorithm and the cutting surface algorithm in solving distributionally robust versions of a few instances from the Stochastic Integer Programming Library, in particular stochastic server location and stochastic multiple binary knapsack problem instances. We also discuss the usefulness of incorporating partial distribution information in two-stage stochastic optimization problems.