1,571 research outputs found
Probability Distributions on Partially Ordered Sets and Network Interdiction Games
This article poses the following problem: Does there exist a probability
distribution over subsets of a finite partially ordered set (poset), such that
a set of constraints involving marginal probabilities of the poset's elements
and maximal chains is satisfied? We present a combinatorial algorithm to
positively resolve this question. The algorithm can be implemented in
polynomial time in the special case where maximal chain probabilities are
affine functions of their elements. This existence problem is relevant for the
equilibrium characterization of a generic strategic interdiction game on a
capacitated flow network. The game involves a routing entity that sends its
flow through the network while facing path transportation costs, and an
interdictor who simultaneously interdicts one or more edges while facing edge
interdiction costs. Using our existence result on posets and strict
complementary slackness in linear programming, we show that the Nash equilibria
of this game can be fully described using primal and dual solutions of a
minimum-cost circulation problem. Our analysis provides a new characterization
of the critical components in the interdiction game. It also leads to a
polynomial-time approach for equilibrium computation
Separable Convex Optimization with Nested Lower and Upper Constraints
We study a convex resource allocation problem in which lower and upper bounds
are imposed on partial sums of allocations. This model is linked to a large
range of applications, including production planning, speed optimization,
stratified sampling, support vector machines, portfolio management, and
telecommunications. We propose an efficient gradient-free divide-and-conquer
algorithm, which uses monotonicity arguments to generate valid bounds from the
recursive calls, and eliminate linking constraints based on the information
from sub-problems. This algorithm does not need strict convexity or
differentiability. It produces an -approximate solution for the
continuous problem in time
and an integer solution in time, where is
the number of decision variables, is the number of constraints, and is
the resource bound. A complexity of is also achieved
for the linear and quadratic cases. These are the best complexities known to
date for this important problem class. Our experimental analyses confirm the
good performance of the method, which produces optimal solutions for problems
with up to 1,000,000 variables in a few seconds. Promising applications to the
support vector ordinal regression problem are also investigated
A Decomposition Algorithm for Nested Resource Allocation Problems
We propose an exact polynomial algorithm for a resource allocation problem
with convex costs and constraints on partial sums of resource consumptions, in
the presence of either continuous or integer variables. No assumption of strict
convexity or differentiability is needed. The method solves a hierarchy of
resource allocation subproblems, whose solutions are used to convert
constraints on sums of resources into bounds for separate variables at higher
levels. The resulting time complexity for the integer problem is , and the complexity of obtaining an -approximate
solution for the continuous case is , being
the number of variables, the number of ascending constraints (such that ), a desired precision, and the total resource. This
algorithm attains the best-known complexity when , and improves it when
. Extensive experimental analyses are conducted with four
recent algorithms on various continuous problems issued from theory and
practice. The proposed method achieves a higher performance than previous
algorithms, addressing all problems with up to one million variables in less
than one minute on a modern computer.Comment: Working Paper -- MIT, 23 page
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