A model of two-settlement electricity markets is introduced, which accounts for flow congestion, demand uncertainty, system contingencies, and market power. We formulate the subgame perfect Nash equilibrium for this model as an equilibrium problem with equilibrium constraints (EPEC), in which each firm solves a mathematical program with equilibrium constraints (MPEC). The model assumes linear demand functions, quadratic generation cost functions, and a lossless DC network, resulting in equilibrium constraints as a parametric linear complementarity problem (LCP). We introduce an iterative procedure for solving this EPEC through repeated application of an MPEC algorithm. This MPEC algorithm is based on solving quadratic programming subproblems and on parametric LCP pivoting. Numerical examples demonstrate the effectiveness of the MPEC and EPEC algorithms and the tractability of the model for realistic-size power systems.Subject classifications: noncooperative games: Cournot equilibrium; electricity market, two settlements; programming: mathematical program with equilibrium constraints, equilibrium problem with equilibrium constraints, linear complementarity problem.
Abstract. We present a new definition of optimality intervals for the parametric right-hand side linear programming (parametric RHS LP) Problem r = min{crxlAx = b + 2b, x > 0}. We then show that an optimality interval consists eifher of a breakpoint or the open interval between two consecutive breakpoints of the continuous piecewise linear convex function ~o(2). As a consequence, the optimality intervals form a partition of the closed interval {2; 1~0(2)1 < oo}. Based on these optimality intervals, we also introduce an algorithm for solving the parametric RHS LP problem which requires an LP solver as a subroutine. If a polynomial-time LP solver is used to implement this subroutine, we obtain a substantial improvement on the complexity of those parametric RHS LP instances which exhibit degeneracy. When the number of breakpoints of q~ (2) is polynomial in terms of the size of the parametric problem, we show that the latter can be solved in polynomial time.
In 1951, Dantzig showed the equivalence of linear programming problems and two-person zerosum games. However, in the description of his reduction from linear programs to zero-sum games, he noted that there was one case in which the reduction does not work. This also led to incomplete proofs of the relationship between the Minimax Theorem of game theory and the Strong Duality Theorem of linear programming. In this note, we fill these gaps.
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