The interior point methods for linear programming problems require solving linear systems in each iteration. Thus, the matrix structure becomes an important issue. For instance, the presence of dense columns in the constraints matrix, makes it crucial to employ specific strategies for solving such linear systems. In this work, we introduce a preconditioner aimed at addressing the linear systems arising from linear programming problems with dense columns. This preconditioner is applied to a modified augmented system to compute the search directions. We show that the resulting linear system remains uniformly bounded with the respect of the iterates as the primal dual interior point method converges to an optimal solution. To validate our approach, we perform computational experiments to linear programming problems with at least one dense column. In comparison with existing methods, our proposal stands out in terms of running time and total number of iteration. The proposed preconditioner successfully solves all the linear systems for the tested linear programming problems showing its robustness.

It is well known that Cournot games involving heterogeneous players with affine and decreasing demand functions and convex quadratic production cost functions can equivalently be cast as a mixed linear complementarity problem or a quadratic optimization problem (QP). Market equilibria can thus be computed using a complementarity solver or, more efficiently, a convex optimization solver. When studying such games, it is common to aggregate the consumers and substitute their inverse demand function to eliminate the market price. We argue, however, that the market price should be treated as an explicit variable that is central to the game. We show that the problem can be reformulated as a fixed-point problem (FPP) for the price only, thereby greatly reducing its dimensionality. The FPP can therefore be solved even more efficiently than the QP and also yields a more accurate solution. The FPP further generalizes to spatial games and games involving multiple commodities. We envision that this could vastly improve solution efficiency and accuracy when incorporated into dynamic or multilevel games. We finally consider a Cournot game across the markets for two commodities that may be converted into each other. The FPP formulation reveals the structure of the joint equilibrium and allows an analytical stability and sensitivity analysis. We find that market imperfections in either of the markets extend to the opposite market and may significantly reduce the welfare gains from market coupling.

A well-known use case for intertemporal multistage stochastic optimization problems is the operation of hydro power plants: this is the type of optimization problem that led to the development of novel techniques such as Stochastic Dual Dynamic Programming (SDDP) (introduced by Pereira and Pinto, 1991), currently used as state of the art for a number of real-world applications. This type of strategy has proven highly effective for solving the central planner's optimization problem (cost minimization/welfare maximization), which can be shown to be equivalent to the Nash equilibrium achieved in a competitive market under certain regularity conditions (as demonstrated by Philpott, Ferris, Wets, 2016). However, the key hypotheses for this equivalence result are rarely valid in practice - in particular, for cascaded hydro power plants, the dependency between the water available for downstream plants and operating choices made at upstream power plants yields an externality, given that this coupling is not remunerated (as explored by Lino, Barroso, et al, 2003).

The main goal of this paper is twofold: firstly, to discuss possible strategies that could be used to "internalize" this externality as part of the electricity market design; and secondly, to present quantitative results for market equilibria involving hydropower cascades with and without the proposed mitigating mechanisms, and a varying number of agents participating in the market. Key strategies for the design of so-called “mitigation mechanisms” to correct for these externalities include vertical consolidation, wholesale water markets, and virtual reservoir mechanisms (see Resende and Cunha, 2022); and the quantitative results obtained can help illustrate the relative magnitudes of the effect of externalities VS the effect of market power for various benchmark systems.

We investigate counterparty credit risk and credit valuation adjustments in
portfolios including derivatives with early-exercise opportunities, under a
netting agreement. We show that credit risk and netting agreements have a
significant impact on the way portfolios are managed (that is, on options'
exercise strategies) and, therefore, on the value of the portfolio and on
the price of counterparty risk. We derive the value of a netted portfolio as
the solution of a zero-sum, finite horizon, discrete-time stochastic game.
We show that this dynamic-game interpretation can be used to determine the
value of the reglementary capital charges required of financial institutions
to cover for counterparty credit risk and we propose a numerical valuation
method. Numerical investigations show that currently used numerical
approaches can grossly misestimate the value of credit valuation adjustments.