This paper provides a review and commentary on the past, present, and future of numerical optimization algorithms in the context of machine learning applications. Through case studies on text classification and the training of deep neural networks, we discuss how optimization problems arise in machine learning and what makes them challenging. A major theme of … Read more
Two approaches to time consistency of risk averse multistage stochastic problems were dis- cussed in the recent literature. In one approach certain properties of the corresponding risk measure are postulated which imply its decomposability. The other approach deals directly with conditional optimality of solutions of the considered problem. The aim of this paper is to … Read more
With greater penetration of renewable generation, the uncertainty faced in electricity markets has increased substantially. Conventionally, generators are assigned a pre-dispatch quantity in advance of real time, based on estimates of uncertain quantities. Expensive real time adjustments then need to be made to ensure demand is met, as uncertainty takes on a realization. We propose … Read more
In recent papers, we have shown that a Lebesgue-Stieltjes optimal control theory forms the foundations for unscented optimal control. In this paper, we further our results by incorporating uncertain mixed state-control constraints in the problem formulation. We show that the integrated Hamiltonian minimization condition resembles a semi-infinite type mathematical programming problem. The resulting computational difficulties … Read more
The bi-objective R&S problem is a special case of the multi-objective simulation optimization problem in which two conflicting objectives are known only through dependent Monte Carlo estimators, the decision space or number of systems is finite, and each system can be sampled to some extent. The solution to the bi-objective R&S problem is a set … Read more
Multi-asset class (MAC) portfolios can be comprised of investments in equities, fixed-income, commodities, foreign-exchange, credit, derivatives, and alternatives such as real-estate and private equity. The return for such {\em non-linear} portfolios is {\em asymmetric} with significant tail risk. The traditional Markowitz Mean-Variance Optimization (MVO) framework, that linearizes all the assets in the portfolio and uses … Read more
Distributionally robust optimization (DRO) is widely used, because it offers a way to overcome the conservativeness of robust optimization without requiring the specificity of stochastic optimization. On the computational side, many practical DRO instances can be equivalently (or approximately) formulated as semidefinite programming (SDP) problems via conic duality of the moment problem. However, despite being … Read more
We study the chance-constrained vehicle routing problem (CCVRP), a version of the vehicle routing problem (VRP) with stochastic demands, where a limit is imposed on the probability that each vehicle’s capacity is exceeded. A distinguishing feature of our proposed methodologies is that they allow correlation between random demands, whereas nearly all existing exact methods for … Read more
Multistage stochastic integer programming (MSIP) combines the difficulty of uncertainty, dynamics, and non-convexity, and constitutes a class of extremely challenging problems. A common formulation for these problems is a dynamic programming formulation involving nested cost-to-go functions. In the linear setting, the cost-to-go functions are convex polyhedral, and decomposition algorithms, such as nested Benders’ decomposition and … Read more
Mixed Integer Dynamic Approximation Scheme (MIDAS) is a new sampling-based algorithm for solving finite-horizon stochastic dynamic programs with monotonic Bellman functions. MIDAS approximates these value functions using step functions, leading to stage problems that are mixed integer programs. We provide a general description of MIDAS, and prove its almost-sure convergence to an epsilon-optimal policy when … Read more