Large-scale constrained convex optimization problems arise in several application domains. First-order methods are good candidates to tackle such problems due to their low iteration complexity and memory requirement. The level-set framework extends the applicability of first-order methods to tackle problems with complicated convex objectives and constraint sets. Current methods based on this framework either rely on the solution of challenging subproblems or do not guarantee a feasible solution, especially if the procedure is terminated before convergence. We develop a level-set method that finds an-relative optimal and feasible solution to a constrained convex optimization problem with a fairly general objective function and set of constraints, maintains a feasible solution at each iteration, and only relies on calls to first-order oracles. We establish the iteration complexity of our approach, also accounting for the smoothness and strong convexity of the objective function and constraints when these properties hold. The dependence of our complexity on is similar to the analogous dependence in the unconstrained setting, which is not known to be true for level-set methods in the literature. Nevertheless, ensuring feasibility is not free. The iteration complexity of our method depends on a condition number, while existing level-set methods that do not guarantee feasibility can avoid such dependence. We numerically validate the usefulness of ensuring a feasible solution path by comparing our approach with an existing level set method on a Neyman-Pearson classification problem.

Least squares Monte Carlo (LSM) is a state-of-the-art approximate dynamic programming approach used in financial engineering and real options to value and manage options with early or multiple exercise opportunities. It is also applicable to capacity investment and inventory/production management problems with demand/supply forecast updates arising in operations and hydropowerreservoir management. LSM has two variants, referred to as regress-now/later (LSMN/L), which compute continuation/value function approximations (C/VFAs). We provide novel numerical evidence for the relative performance of these methods applied to energy swing and storage options, two typical real options, using a common price evolution model. LSMN/L estimate C/VFAs that yield equally accurate (near optimal) and precise lower and dual (upper) bounds on the value of these real options. Estimating the LSMN/L C/VFAs and their associated lower bounds takes similar computational effort. In contrast, the estimation of a dual bound using the LSML VFA instead of the LSMN CFA takes seconds rather than minutes or hours. This finding suggests the use of LSML in lieu of LSMN when estimating dual bounds on the value of early or multiple exercise options, as well as of related capacity investment and inventory/production policies.

The real option management of commodity conversion assets gives rise to intractable Markov decision processes (MDPs). This is due primarily to the high dimensionality of a commodity forward curve, which is part of the MDP state when using high dimensional models of the evolution of this curve, as commonly done in practice. Focusing on commodity storage, we develop a novel approximate dynamic programming methodology that hinges on the relaxation of approximate linear programs (ALPs) obtained using value function approximations based on reducing the number of futures prices that are part of the MDP state. We derive equivalent approximate dynamic programs (ADPs) for a class of these ALPs, also subsuming a known ADP. We obtain two new ADPs, the value functions of which induce feasible policies for the original MDP, and lower and upper bounds, estimated via Monte Carlo simulation, on the value of an optimal policy of this MDP. We investigate the performance of our ADPs on existing natural gas instances and new crude oil instances. Our approach has potential relevance for the approximate solution of MDPs that arise in the real option management of other commodity conversion assets, as well as the valuation and management of real and financial options that depend on forward curve dynamics.

The real option management of commodity conversion assets gives rise to intractable Markov decision processes (MDPs). This is due primarily to the high dimensionality of a commodity forward curve, which is part of the MDP state when using high dimensional models of the evolution of this curve, as commonly done in practice. Focusing on commodity storage, we develop a novel approximate dynamic programming methodology that hinges on the relaxation of approximate linear programs (ALPs) obtained using value function approximations based on reducing the number of futures prices that are part of the MDP state. We derive equivalent approximate dynamic programs (ADPs) for a class of these ALPs, also subsuming a known ADP. We obtain two new ADPs, the value functions of which induce feasible policies for the original MDP, and lower and upper bounds, estimated via Monte Carlo simulation, on the value of an optimal policy of this MDP. We investigate the performance of our ADPs on existing natural gas instances and new crude oil instances. Our approach has potential relevance for the approximate solution of MDPs that arise in the real option management of other commodity conversion assets, as well as the valuation and management of real and financial options that depend on forward curve dynamics.

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