We present a new primal-dual algorithm for computing the value of the Lagrangian dual of a stochastic mixed-integer program (SMIP) formed by relaxing its nonanticipativity constraints. This dual is widely used in decomposition methods for the solution of SMIPs. The algorithm relies on the well-known progressive hedging method, * but unlike previous progressive hedging approaches for SMIP, our algorithm can be shown to converge to the optimal Lagrangian dual value. The key improvement in the new algorithm is an inner loop of optimized linearization steps, similar to those taken in the classical Frank-Wolfe method. Numerical results demonstrate that our new algorithm empirically outperforms the standard implementation of progressive hedging for obtaining bounds in SMIP.

We give exact criteria for the -divisibility of the -regular partition function b (n) for ∈ {5, 7, 11}. These criteria are found using the theory of complex multiplication. In each case the first criterion given corresponds to the Ramanujan congruence modulo for the unrestricted partition function, and the second is a condition given by J.-P. Serre for the vanishing of the coefficients of ∞ m=1 (1 − q m ) −1 .

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This work considers the impact of thermal behavior in battery design. The cell performance worsens when the operating temperature falls outside of the ideal range, and evenness of cell temperatures is sought to prevent cell electrical unbalance which may lead to performance fading and premature failure. The heat transfer between the cells and the coolant depends on the cell packaging and layout. A multi-objective optimization model is posed whose Pareto efficient designs minimize cell temperature deviations while maintaining evenness of temperature distribution. The special characteristics of the battery design problem (comparable objectives, anonymity and Pigou–Dalton principle of transfers) make it suitable for the application of the equitability preference, which is a refinement of the Pareto optimality that has not been used in engineering design. The proposed approach based on equitability is applied to compute the spacing of the cylindrical cells in a battery module that yields an optimal thermal behavior.