31,647 research outputs found
Approximating optimization problems over convex functions
Many problems of theoretical and practical interest involve finding an
optimum over a family of convex functions. For instance, finding the projection
on the convex functions in , and optimizing functionals arising
from some problems in economics.
In the continuous setting and assuming smoothness, the convexity constraints
may be given locally by asking the Hessian matrix to be positive semidefinite,
but in making discrete approximations two difficulties arise: the continuous
solutions may be not smooth, and functions with positive semidefinite discrete
Hessian need not be convex in a discrete sense.
Previous work has concentrated on non-local descriptions of convexity, making
the number of constraints to grow super-linearly with the number of nodes even
in dimension 2, and these descriptions are very difficult to extend to higher
dimensions.
In this paper we propose a finite difference approximation using positive
semidefinite programs and discrete Hessians, and prove convergence under very
general conditions, even when the continuous solution is not smooth, working on
any dimension, and requiring a linear number of constraints in the number of
nodes.
Using semidefinite programming codes, we show concrete examples of
approximations to problems in two and three dimensions
Stability and Error Analysis for Optimization and Generalized Equations
Stability and error analysis remain challenging for problems that lack
regularity properties near solutions, are subject to large perturbations, and
might be infinite dimensional. We consider nonconvex optimization and
generalized equations defined on metric spaces and develop bounds on solution
errors using the truncated Hausdorff distance applied to graphs and epigraphs
of the underlying set-valued mappings and functions. In the process, we extend
the calculus of such distances to cover compositions and other constructions
that arise in nonconvex problems. The results are applied to constrained
problems with feasible sets that might have empty interiors, solution of KKT
systems, and optimality conditions for difference-of-convex functions and
composite functions
Approximate Dynamic Programming via Sum of Squares Programming
We describe an approximate dynamic programming method for stochastic control
problems on infinite state and input spaces. The optimal value function is
approximated by a linear combination of basis functions with coefficients as
decision variables. By relaxing the Bellman equation to an inequality, one
obtains a linear program in the basis coefficients with an infinite set of
constraints. We show that a recently introduced method, which obtains convex
quadratic value function approximations, can be extended to higher order
polynomial approximations via sum of squares programming techniques. An
approximate value function can then be computed offline by solving a
semidefinite program, without having to sample the infinite constraint. The
policy is evaluated online by solving a polynomial optimization problem, which
also turns out to be convex in some cases. We experimentally validate the
method on an autonomous helicopter testbed using a 10-dimensional helicopter
model.Comment: 7 pages, 5 figures. Submitted to the 2013 European Control
Conference, Zurich, Switzerlan
Approximations of Semicontinuous Functions with Applications to Stochastic Optimization and Statistical Estimation
Upper semicontinuous (usc) functions arise in the analysis of maximization
problems, distributionally robust optimization, and function identification,
which includes many problems of nonparametric statistics. We establish that
every usc function is the limit of a hypo-converging sequence of piecewise
affine functions of the difference-of-max type and illustrate resulting
algorithmic possibilities in the context of approximate solution of
infinite-dimensional optimization problems. In an effort to quantify the ease
with which classes of usc functions can be approximated by finite collections,
we provide upper and lower bounds on covering numbers for bounded sets of usc
functions under the Attouch-Wets distance. The result is applied in the context
of stochastic optimization problems defined over spaces of usc functions. We
establish confidence regions for optimal solutions based on sample average
approximations and examine the accompanying rates of convergence. Examples from
nonparametric statistics illustrate the results
Data Filtering for Cluster Analysis by -Norm Regularization
A data filtering method for cluster analysis is proposed, based on minimizing
a least squares function with a weighted -norm penalty. To overcome the
discontinuity of the objective function, smooth non-convex functions are
employed to approximate the -norm. The convergence of the global
minimum points of the approximating problems towards global minimum points of
the original problem is stated. The proposed method also exploits a suitable
technique to choose the penalty parameter. Numerical results on synthetic and
real data sets are finally provided, showing how some existing clustering
methods can take advantages from the proposed filtering strategy.Comment: Optimization Letters (2017
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