15,051 research outputs found
An almost cyclic 2-coordinate descent method for singly linearly constrained problems
A block decomposition method is proposed for minimizing a (possibly
non-convex) continuously differentiable function subject to one linear equality
constraint and simple bounds on the variables. The proposed method iteratively
selects a pair of coordinates according to an almost cyclic strategy that does
not use first-order information, allowing us not to compute the whole gradient
of the objective function during the algorithm. Using first-order search
directions to update each pair of coordinates, global convergence to stationary
points is established for different choices of the stepsize under an
appropriate assumption on the level set. In particular, both inexact and exact
line search strategies are analyzed. Further, linear convergence rate is proved
under standard additional assumptions. Numerical results are finally provided
to show the effectiveness of the proposed method.Comment: Computational Optimization and Application
Conditional Gradient Algorithms for Rank-One Matrix Approximations with a Sparsity Constraint
The sparsity constrained rank-one matrix approximation problem is a difficult
mathematical optimization problem which arises in a wide array of useful
applications in engineering, machine learning and statistics, and the design of
algorithms for this problem has attracted intensive research activities. We
introduce an algorithmic framework, called ConGradU, that unifies a variety of
seemingly different algorithms that have been derived from disparate
approaches, and allows for deriving new schemes. Building on the old and
well-known conditional gradient algorithm, ConGradU is a simplified version
with unit step size and yields a generic algorithm which either is given by an
analytic formula or requires a very low computational complexity. Mathematical
properties are systematically developed and numerical experiments are given.Comment: Minor changes. Final version. To appear in SIAM Revie
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Using EPECs to model bilevel games in restructured electricity markets with locational prices
CWPE0619 (EPRG0602) Xinmin Hu and Daniel Ralph (Feb 2006) Using EPECs to model bilevel games in restructured electricity markets with locational prices We study a bilevel noncooperative game-theoretic model of electricity markets with locational marginal prices. Each player faces a bilevel optimization problem that we remodel as a mathematical program with equilibrium constraints, MPEC. This gives an EPEC, equilibrium problem with equilibrium constraints. We establish sufficient conditions for existence of pure strategy Nash equilibria for this class of bilevel games and give some applications. We show by examples the effect of network transmission limits, i.e. congestion, on existence of equilibria. Then we study, for more general EPECs, the weaker pure strategy concepts of local Nash and Nash stationary equilibria. We model the latter via complementarity problems, CPs. Finally, we present numerical examples of methods that attempt to find local Nash or Nash stationary equilibria of randomly generated electricity market games. The CP solver PATH is found to be rather effective in this context
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