257,750 research outputs found
Global Optimization for Value Function Approximation
Existing value function approximation methods have been successfully used in
many applications, but they often lack useful a priori error bounds. We propose
a new approximate bilinear programming formulation of value function
approximation, which employs global optimization. The formulation provides
strong a priori guarantees on both robust and expected policy loss by
minimizing specific norms of the Bellman residual. Solving a bilinear program
optimally is NP-hard, but this is unavoidable because the Bellman-residual
minimization itself is NP-hard. We describe and analyze both optimal and
approximate algorithms for solving bilinear programs. The analysis shows that
this algorithm offers a convergent generalization of approximate policy
iteration. We also briefly analyze the behavior of bilinear programming
algorithms under incomplete samples. Finally, we demonstrate that the proposed
approach can consistently minimize the Bellman residual on simple benchmark
problems
Locally Adaptive Optimization: Adaptive Seeding for Monotone Submodular Functions
The Adaptive Seeding problem is an algorithmic challenge motivated by
influence maximization in social networks: One seeks to select among certain
accessible nodes in a network, and then select, adaptively, among neighbors of
those nodes as they become accessible in order to maximize a global objective
function. More generally, adaptive seeding is a stochastic optimization
framework where the choices in the first stage affect the realizations in the
second stage, over which we aim to optimize.
Our main result is a -approximation for the adaptive seeding
problem for any monotone submodular function. While adaptive policies are often
approximated via non-adaptive policies, our algorithm is based on a novel
method we call \emph{locally-adaptive} policies. These policies combine a
non-adaptive global structure, with local adaptive optimizations. This method
enables the -approximation for general monotone submodular functions
and circumvents some of the impossibilities associated with non-adaptive
policies.
We also introduce a fundamental problem in submodular optimization that may
be of independent interest: given a ground set of elements where every element
appears with some small probability, find a set of expected size at most
that has the highest expected value over the realization of the elements. We
show a surprising result: there are classes of monotone submodular functions
(including coverage) that can be approximated almost optimally as the
probability vanishes. For general monotone submodular functions we show via a
reduction from \textsc{Planted-Clique} that approximations for this problem are
not likely to be obtainable. This optimization problem is an important tool for
adaptive seeding via non-adaptive policies, and its hardness motivates the
introduction of \emph{locally-adaptive} policies we use in the main result
Theory and applications of the Sum-Of-Squares technique
The Sum-of-Squares (SOS) approximation method is a technique used in
optimization problems to derive lower bounds to the optimal value of an
objective function. By representing the objective function as a sum of squares
in a feature space, the SOS method transforms non-convex global optimization
problems into solvable semidefinite programs. This note presents an overview of
the SOS method. We start with its application in finite-dimensional feature
spaces and, subsequently, we extend it to infinite-dimensional feature spaces
using kernels (k-SOS). Additionally, we highlight the utilization of SOS for
estimating some relevant quantities in information theory, including the
log-partition function.Comment: 19 pages, 4 figure
Reservoir Flooding Optimization by Control Polynomial Approximations
In this dissertation, we provide novel parametrization procedures for water-flooding
production optimization problems, using polynomial approximation techniques. The methods project the original infinite dimensional controls space into a polynomial subspace. Our contribution includes new parameterization formulations using natural polynomials, orthogonal Chebyshev polynomials and Cubic spline interpolation.
We show that the proposed methods are well suited for black-box approach with
stochastic global-search method as they tend to produce smooth control trajectories, while reducing the solution space size. We demonstrate their efficiency on synthetic two-dimensional problems and on a realistic 3-dimensional problem.
By contributing with a new adjoint method formulation for polynomial approximation,
we implemented the methods also with gradient-based algorithms. In addition to fine-scale simulation, we also performed reduced order modeling, where we demonstrated a synergistic effect when combining polynomial approximation with model order reduction, that leads to faster optimization with higher gains in terms of Net Present Value.
Finally, we performed gradient-based optimization under uncertainty. We proposed
a new multi-objective function with three components, one that maximizes the expected
value of all realizations, and two that maximize the averages of distribution tails from
both sides. The new objective provides decision makers with the flexibility to choose the
amount of risk they are willing to take, while deciding on production strategy or performing reserves estimation (P10;P50;P90)
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