11,952 research outputs found
Inexactness of the Hydro-Thermal Coordination Semidefinite Relaxation
Hydro-thermal coordination is the problem of determining the optimal economic
dispatch of hydro and thermal power plants over time. The physics of
hydroelectricity generation is commonly simplified in the literature to account
for its fundamentally nonlinear nature. Advances in convex relaxation theory
have allowed the advent of Shor's semidefinite programming (SDP) relaxations of
quadratic models of the problem. This paper shows how a recently published SDP
relaxation is only exact if a very strict condition regarding turbine
efficiency is observed, failing otherwise. It further proposes the use of a set
of convex envelopes as a strategy to successfully obtain a stricter lower bound
of the optimal solution. This strategy is combined with a standard iterative
convex-concave procedure to recover a stationary point of the original
non-convex problem.Comment: Submitted to IEEE PES General Meeting 201
Scheduling for a Processor Sharing System with Linear Slowdown
We consider the problem of scheduling arrivals to a congestion system with a
finite number of users having identical deterministic demand sizes. The
congestion is of the processor sharing type in the sense that all users in the
system at any given time are served simultaneously. However, in contrast to
classical processor sharing congestion models, the processing slowdown is
proportional to the number of users in the system at any time. That is, the
rate of service experienced by all users is linearly decreasing with the number
of users. For each user there is an ideal departure time (due date). A
centralized scheduling goal is then to select arrival times so as to minimize
the total penalty due to deviations from ideal times weighted with sojourn
times. Each deviation is assumed quadratic, or more generally convex. But due
to the dynamics of the system, the scheduling objective function is non-convex.
Specifically, the system objective function is a non-smooth piecewise convex
function. Nevertheless, we are able to leverage the structure of the problem to
derive an algorithm that finds the global optimum in a (large but) finite
number of steps, each involving the solution of a constrained convex program.
Further, we put forward several heuristics. The first is the traversal of
neighbouring constrained convex programming problems, that is guaranteed to
reach a local minimum of the centralized problem. This is a form of a "local
search", where we use the problem structure in a novel manner. The second is a
one-coordinate "global search", used in coordinate pivot iteration. We then
merge these two heuristics into a unified "local-global" heuristic, and
numerically illustrate the effectiveness of this heuristic
Lift-and-Round to Improve Weighted Completion Time on Unrelated Machines
We consider the problem of scheduling jobs on unrelated machines so as to
minimize the sum of weighted completion times. Our main result is a
-approximation algorithm for some fixed , improving upon the
long-standing bound of 3/2 (independently due to Skutella, Journal of the ACM,
2001, and Sethuraman & Squillante, SODA, 1999). To do this, we first introduce
a new lift-and-project based SDP relaxation for the problem. This is necessary
as the previous convex programming relaxations have an integrality gap of
. Second, we give a new general bipartite-rounding procedure that produces
an assignment with certain strong negative correlation properties.Comment: 21 pages, 4 figure
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