Sample-path optimization is a simulation-based method for solving optimization problems that azise in the study of complex stochastic systems. In this paper we broaden its applicability to include the solution of stochastic vaziational inequalitiea. This formulation can model equilibrium phenomena in physics, economics, and operations reseazch. We describe the method, provide general conditions for convergence, and present numerical results of an application of the method to a stochastic economic equilibrium model of the European natural gas market. We alsa point out some current ]imitations of the method and indicate azeas in which research might help to remove those limitations.

Inspired by previous works on approximations of optimization problems and recent papers on the approximation of Walrasian and Nash equilibria and on stochastic variational inequalities, the present paper investigates the approximation of Nash equilibria and clarifies the conditions required for the convergence of the approximate equilibria via a direct approach, a variational approach, and an optimization approach. Besides directly addressing the issue of convergence of Nash equilibria via approximation, our investigation leads to a deeper understanding of various notions of functional convergence and their interconnections; more importantly, the investigation yields improved conditions for convergence of the approximate Nash equilibria via the variational approach. An illustrative application of our results to the It is with great pleasure that we dedicate this paper to our esteemed colleague and mentor, Professor Steve Robinson, on the occasion of his 65th birthday. Steve's many seminal works have deeply influenced our research; the present paper is an example of his influence.

In competitive electricity markets, markets designs based on power exchanges where supply bidding (barring demand-side bidding) is at the sole short run marginal cost may not guarantee resource adequacy. As alternative ways to remedy the resource adequacy problem, we focus on three different market designs in detail when demand is inelastic, namely an energy-only market with VOLL pricing (or a price cap), an additional capacity market, and operating-reserve pricing. We also discuss demand-side bidding (i.e., a price responsive demand) which can be seen as a categorically different alternative to remedy the resource adequacy problem. We consider a perfectly competitive market consisting of three types of agents: generators, a transmission system operator, and consumers; all agents are assumed to have no market power. For each market design, we model and analyze capacity investment choices of firms using a two-stage game where generation capacities are installed in the first stage and generation takes place in future spot markets at the second stage. When future spot market conditions are assumed to be known a priori (i.e., deterministic demand case), we show that all of these two-stage models with different market mechanisms, except operating-reserve pricing, can be cast as single optimization problems. When future spot market conditions are not known in advance (i.e., under demand uncertainty), we essentially have a two-stage stochastic game. Interestingly, an equilibrium point of this stochastic game can be found by solving a two-stage stochastic program, in case of all of the market mechanisms except operating-reserve pricing. In case of operatingreserve pricing, while the formulation of an equivalent deterministic or stochastic optimization problem is possible when operating-reserves are based on observed demand, this simplicity is lost when operatingreserves are based on installed capacities. We generalize these results for other uncertain parameters in spot markets such as fuel costs and transmission capacities. Finally, we illustrate how all these models can be numerically tackled and present numerical experiments. In our numerical experiments, we observe that uncertainty of demand leads to higher total generation capacity expansion and a broader mix of technologies compared to the investment decisions assuming average demand levels. Furthermore for the same VOLL (or price cap) level and under the assumptions of random demand with finite support and no forced outages, energy-only markets with VOLL pricing tend to lead to total generation capacity below the peak load with a

No abstract

This paper shows how to apply a variant of samplepath optimization to solve stochastic variational inequalities, including as a special case finding a zero of a gradient. We give a new set of sufficient conditions for almost-sure convergence of the method, and exhibit bounds on the error of the resulting approximate solution. We also illustrate the application of this method by using it to price an American call option on a dividend-paying stock.