2,264 research outputs found
A partial differential equation for the strictly quasiconvex envelope
In a series of papers Barron, Goebel, and Jensen studied Partial Differential
Equations (PDE)s for quasiconvex (QC) functions \cite{barron2012functions,
barron2012quasiconvex,barron2013quasiconvex,barron2013uniqueness}. To overcome
the lack of uniqueness for the QC PDE, they introduced a regularization: a PDE
for \e-robust QC functions, which is well-posed. Building on this work, we
introduce a stronger regularization which is amenable to numerical
approximation. We build convergent finite difference approximations, comparing
the QC envelope and the two regularization. Solutions of this PDE are strictly
convex, and smoother than the robust-QC functions.Comment: 20 pages, 6 figures, 1 tabl
New Formulation and Strong MISOCP Relaxations for AC Optimal Transmission Switching Problem
As the modern transmission control and relay technologies evolve,
transmission line switching has become an important option in power system
operators' toolkits to reduce operational cost and improve system reliability.
Most recent research has relied on the DC approximation of the power flow model
in the optimal transmission switching problem. However, it is known that DC
approximation may lead to inaccurate flow solutions and also overlook stability
issues. In this paper, we focus on the optimal transmission switching problem
with the full AC power flow model, abbreviated as AC OTS. We propose a new
exact formulation for AC OTS and its mixed-integer second-order conic
programming (MISOCP) relaxation. We improve this relaxation via several types
of strong valid inequalities inspired by the recent development for the closely
related AC Optimal Power Flow (AC OPF) problem. We also propose a practical
algorithm to obtain high quality feasible solutions for the AC OTS problem.
Extensive computational experiments show that the proposed formulation and
algorithms efficiently solve IEEE standard and congested instances and lead to
significant cost benefits with provably tight bounds
Risk Minimization, Regret Minimization and Progressive Hedging Algorithms
This paper begins with a study on the dual representations of risk and regret
measures and their impact on modeling multistage decision making under
uncertainty. A relationship between risk envelopes and regret envelopes is
established by using the Lagrangian duality theory. Such a relationship opens a
door to a decomposition scheme, called progressive hedging, for solving
multistage risk minimization and regret minimization problems. In particular,
the classical progressive hedging algorithm is modified in order to handle a
new class of linkage constraints that arises from reformulations and other
applications of risk and regret minimization problems. Numerical results are
provided to show the efficiency of the progressive hedging algorithms.Comment: 21 pages, 2 figure
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