1,288 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
A Scalable Algorithm For Sparse Portfolio Selection
The sparse portfolio selection problem is one of the most famous and
frequently-studied problems in the optimization and financial economics
literatures. In a universe of risky assets, the goal is to construct a
portfolio with maximal expected return and minimum variance, subject to an
upper bound on the number of positions, linear inequalities and minimum
investment constraints. Existing certifiably optimal approaches to this problem
do not converge within a practical amount of time at real world problem sizes
with more than 400 securities. In this paper, we propose a more scalable
approach. By imposing a ridge regularization term, we reformulate the problem
as a convex binary optimization problem, which is solvable via an efficient
outer-approximation procedure. We propose various techniques for improving the
performance of the procedure, including a heuristic which supplies high-quality
warm-starts, a preprocessing technique for decreasing the gap at the root node,
and an analytic technique for strengthening our cuts. We also study the
problem's Boolean relaxation, establish that it is second-order-cone
representable, and supply a sufficient condition for its tightness. In
numerical experiments, we establish that the outer-approximation procedure
gives rise to dramatic speedups for sparse portfolio selection problems.Comment: Submitted to INFORMS Journal on Computin
Generic optimality conditions for semi-algebraic convex programs
We consider linear optimization over a nonempty convex semi-algebraic feasible region F. Semidefinite programming is an example. If F is compact, then for almost every linear objective there is a unique optimal solution, lying on a unique \active" manifold, around which F is \partly smooth", and the second-order sufficient conditions hold. Perturbing the objective results in smooth variation of the optimal solution. The active manifold consists, locally, of these perturbed optimal solutions; it is independent of the representation of F, and is eventually identified by a variety of iterative algorithms such as proximal and projected gradient schemes. These results extend to unbounded sets F
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