4,466 research outputs found
Theory and Applications of Robust Optimization
In this paper we survey the primary research, both theoretical and applied,
in the area of Robust Optimization (RO). Our focus is on the computational
attractiveness of RO approaches, as well as the modeling power and broad
applicability of the methodology. In addition to surveying prominent
theoretical results of RO, we also present some recent results linking RO to
adaptable models for multi-stage decision-making problems. Finally, we
highlight applications of RO across a wide spectrum of domains, including
finance, statistics, learning, and various areas of engineering.Comment: 50 page
Robustness and Regularization of Support Vector Machines
We consider regularized support vector machines (SVMs) and show that they are
precisely equivalent to a new robust optimization formulation. We show that
this equivalence of robust optimization and regularization has implications for
both algorithms, and analysis. In terms of algorithms, the equivalence suggests
more general SVM-like algorithms for classification that explicitly build in
protection to noise, and at the same time control overfitting. On the analysis
front, the equivalence of robustness and regularization, provides a robust
optimization interpretation for the success of regularized SVMs. We use the
this new robustness interpretation of SVMs to give a new proof of consistency
of (kernelized) SVMs, thus establishing robustness as the reason regularized
SVMs generalize well
New Dependencies of Hierarchies in Polynomial Optimization
We compare four key hierarchies for solving Constrained Polynomial
Optimization Problems (CPOP): Sum of Squares (SOS), Sum of Diagonally Dominant
Polynomials (SDSOS), Sum of Nonnegative Circuits (SONC), and the Sherali Adams
(SA) hierarchies. We prove a collection of dependencies among these hierarchies
both for general CPOPs and for optimization problems on the Boolean hypercube.
Key results include for the general case that the SONC and SOS hierarchy are
polynomially incomparable, while SDSOS is contained in SONC. A direct
consequence is the non-existence of a Putinar-like Positivstellensatz for
SDSOS. On the Boolean hypercube, we show as a main result that Schm\"udgen-like
versions of the hierarchies SDSOS*, SONC*, and SA* are polynomially equivalent.
Moreover, we show that SA* is contained in any Schm\"udgen-like hierarchy that
provides a O(n) degree bound.Comment: 26 pages, 4 figure
Mixed-integer convex representability
Motivated by recent advances in solution methods for mixed-integer convex
optimization (MICP), we study the fundamental and open question of which sets
can be represented exactly as feasible regions of MICP problems. We establish
several results in this direction, including the first complete
characterization for the mixed-binary case and a simple necessary condition for
the general case. We use the latter to derive the first non-representability
results for various non-convex sets such as the set of rank-1 matrices and the
set of prime numbers. Finally, in correspondence with the seminal work on
mixed-integer linear representability by Jeroslow and Lowe, we study the
representability question under rationality assumptions. Under these
rationality assumptions, we establish that representable sets obey strong
regularity properties such as periodicity, and we provide a complete
characterization of representable subsets of the natural numbers and of
representable compact sets. Interestingly, in the case of subsets of natural
numbers, our results provide a clear separation between the mathematical
modeling power of mixed-integer linear and mixed-integer convex optimization.
In the case of compact sets, our results imply that using unbounded integer
variables is necessary only for modeling unbounded sets
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