Proximal Decomposition Methods with Applications to Electricity Production Problems

L.A. PARENTE; P.A. LOTITO; A.J. RUBIALES; M.V. SOLODOV

Valparaíso

Workshop; III LATIN AMERICAN WORKSHOP ON OPTIMIZATION AND CONTROL; 2012

Universidad Técnica Federico Santa Maria

In this talk, the problem of finding a zero of a maximal monotone operator $T:R^n ightrightarrowsR^n$, is considered. As it is well known, a wide variety of problems such as convex optimization, min-max problems, and monotone variational inequalities over convex sets, fall within this general framework. Our central interest is the situation when the given operator $T$ has some separable structure, so decomposition techniques come into play. A variable metric hybrid proximal decomposition method (VMHPDM) was introduced in [Lotito et al.2009], allowing the use of variable metric in subproblems along the lines of [Parente et al.2008]. This new technique turns to be rather general and includes as special cases a number of well known schemes. Global convergence and local linear rate of the convergence are established under standard assumptions. In [Parente et al.2011], VMHPDM is used for computing a solution of a short-term hydrothermal scheduling problem. An oligopolistic electricity market composed of two types of power generation units, thermal and hydroelectric, with a bounded capacity transmission net, is considered. The Nash equilibrium analytic condition is stated as a variational inclusion and it is shown that the associated operator has a structure suitable for decomposition. The application of VMHPDM is illustrated in several examples, in particular, the Transcomahue system, wich is a sector of the Argentine Interconnected System. Numerical results for each example are presented.