Olivier Huber

University of Wisconsin-Madison

January 13, 2016, 11:00, Room GC B2 424 (click here for the map)

Optimization of the sum of a convex surrogate and quadratic objective

We consider a convex optimization problem where the objective function is the sum f(x) + g(y) and the coupling between the variables x and y is at the constraint level. We focus on the case where g is not available in closed form and can only be evaluated at a given point by running a long simulation process. The results of interest are prices formed from the gradient of g. It is assumed that the function g is convex or can be (reasonably) approximated by a convex one. We choose to use an approximation of g defined as a pointwise supremum over a family of piecewise affine functions. This part of the procedure is carried out offline, and uses evaluations of g to define the approximation from its epigraph. We report on using the Moreau-Yosida regularization on our approximation function to return a smoothed value of the gradient that reduces the volatility in the prices. We outline some results in the context of a reserve energy market planning problem.

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