12,921 research outputs found
On the Polytope Escape Problem for Continuous Linear Dynamical Systems
The Polyhedral Escape Problem for continuous linear dynamical systems
consists of deciding, given an affine function and a convex polyhedron ,
whether, for some initial point in , the
trajectory of the unique solution to the differential equation
,
, is entirely contained in .
We show that this problem is decidable, by reducing it in polynomial time to
the decision version of linear programming with real algebraic coefficients,
thus placing it in , which lies between NP and PSPACE. Our
algorithm makes use of spectral techniques and relies among others on tools
from Diophantine approximation.Comment: Accepted to HSCC 201
Hamiltonian cycles and subsets of discounted occupational measures
We study a certain polytope arising from embedding the Hamiltonian cycle
problem in a discounted Markov decision process. The Hamiltonian cycle problem
can be reduced to finding particular extreme points of a certain polytope
associated with the input graph. This polytope is a subset of the space of
discounted occupational measures. We characterize the feasible bases of the
polytope for a general input graph , and determine the expected numbers of
different types of feasible bases when the underlying graph is random. We
utilize these results to demonstrate that augmenting certain additional
constraints to reduce the polyhedral domain can eliminate a large number of
feasible bases that do not correspond to Hamiltonian cycles. Finally, we
develop a random walk algorithm on the feasible bases of the reduced polytope
and present some numerical results. We conclude with a conjecture on the
feasible bases of the reduced polytope.Comment: revised based on referees comment
Optimization with multivariate conditional value-at-risk constraints
For many decision making problems under uncertainty, it is crucial to develop risk-averse models and specify the decision makers' risk preferences based on multiple stochastic performance measures (or criteria). Incorporating such multivariate preference rules into optimization models is a fairly recent research area. Existing studies focus on extending univariate stochastic dominance rules to the multivariate case. However, enforcing multivariate stochastic dominance constraints can often be overly conservative in practice.
As an alternative, we focus on the widely-applied risk measure conditional value-at-risk (CVaR), introduce a multivariate CVaR relation, and develop a novel optimization model with multivariate CVaR constraints based on polyhedral scalarization. To solve such problems for finite probability spaces we develop a cut generation algorithm, where each cut is obtained by solving a mixed integer problem. We show that a multivariate CVaR constraint reduces to
finitely many univariate CVaR constraints, which proves the finite convergence of our algorithm. We also show that our results can be naturally extended to a wider class of coherent risk measures. The proposed approach provides a flexible, and computationally tractable way of modeling preferences in stochastic multi-criteria decision making. We conduct a computational study for a budget allocation problem to illustrate the effect of enforcing multivariate CVaR constraints and demonstrate the computational performance of the
proposed solution methods
Optimization with multivariate conditional value-at-risk constraints
For many decision making problems under uncertainty, it is crucial to develop risk-averse models and specify the decision makers' risk preferences based on multiple stochastic performance measures (or criteria). Incorporating such multivariate preference rules into optimization models is a fairly recent research area. Existing studies focus on extending univariate stochastic dominance rules to the multivariate case. However, enforcing multivariate stochastic dominance constraints can often be overly conservative in practice.
As an alternative, we focus on the widely-applied risk measure conditional value-at-risk (CVaR), introduce a multivariate CVaR relation, and develop a novel optimization model with multivariate CVaR constraints based on polyhedral scalarization. To solve such problems for finite probability spaces we develop a cut generation algorithm, where each cut is obtained by solving a mixed integer problem. We show that a multivariate CVaR constraint reduces to
finitely many univariate CVaR constraints, which proves the finite convergence of our algorithm. We also show that our results can be naturally extended to a wider class of coherent risk measures. The proposed approach provides a flexible, and computationally tractable way of modeling preferences in stochastic multi-criteria decision making. We conduct a computational study for a budget allocation problem to illustrate the effect of enforcing multivariate CVaR constraints and demonstrate the computational performance of the
proposed solution methods
Symbol-Level Precoding Design for Max-Min SINR in Multiuser MISO Broadcast Channels
In this paper, we address the symbol level precoding (SLP) design problem
under max-min SINR criterion in the downlink of multiuser multiple-input
single-output (MISO) channels. First, we show that the distance preserving
constructive interference regions (DPCIR) are always polyhedral angles (shifted
pointed cones) for any given constellation point with unbounded decision
region. Then we prove that any signal in a given unbounded DPCIR has a norm
larger than the norm of the corresponding vertex if and only if the convex hull
of the constellation contains the origin. Using these properties, we show that
the power of the noiseless received signal lying on an unbounded DPCIR is an
strictly increasing function of two parameters. This allows us to reformulate
the originally non-convex SLP max-min SINR as a convex optimization problem. We
discuss the loss due to our proposed convex reformulation and provide some
simulation results.Comment: Submitted to SPAWC 2018, 7 pages, 2 figure
Playing Billiard in Version Space
A ray-tracing method inspired by ergodic billiards is used to estimate the
theoretically best decision rule for a set of linear separable examples. While
the Bayes-optimum requires a majority decision over all Perceptrons separating
the example set, the problem considered here corresponds to finding the single
Perceptron with best average generalization probability. For randomly
distributed examples the billiard estimate agrees with known analytic results.
In real-life classification problems the generalization error is consistently
reduced compared to the maximal stability Perceptron.Comment: uuencoded, gzipped PostScript file, 127576 bytes To recover 1) save
file as bayes.uue. Then 2) uudecode bayes.uue and 3) gunzip bayes.ps.g
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