3,423 research outputs found
Solving variational inequalities defined on a domain with infinitely many linear constraints
We study a variational inequality problem whose domain is defined by infinitely many linear inequalities. A discretization method and an analytic center based inexact cutting plane method are proposed. Under proper assumptions, the convergence results for both methods are given. We also provide numerical examples to illustrate the proposed method
On the Efficient Solution of Variational Inequalities; Complexity and Computational Efficiency
In this paper we combine ideas from cutting plane and interior point methods in order to solve variational inequality problems efficiently. In particular, we introduce a general framework that incorporates nonlinear as well as linear "smarter" cuts. These cuts utilize second order information on the problem through the use of a gap function. We establish convergence as well as complexity results for this framework. Moreover, in order to devise more practical methods, we consider an affine scaling method as it applies to symmetric, monotone variationalinequality problems and demonstrate its convergence. Finally, in order to further improve the computational efficiency of the methods in this paper, we combine the cutting plane approach with the affine scaling approach
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
A Copositive Framework for Analysis of Hybrid Ising-Classical Algorithms
Recent years have seen significant advances in quantum/quantum-inspired
technologies capable of approximately searching for the ground state of Ising
spin Hamiltonians. The promise of leveraging such technologies to accelerate
the solution of difficult optimization problems has spurred an increased
interest in exploring methods to integrate Ising problems as part of their
solution process, with existing approaches ranging from direct transcription to
hybrid quantum-classical approaches rooted in existing optimization algorithms.
While it is widely acknowledged that quantum computers should augment classical
computers, rather than replace them entirely, comparatively little attention
has been directed toward deriving analytical characterizations of their
interactions. In this paper, we present a formal analysis of hybrid algorithms
in the context of solving mixed-binary quadratic programs (MBQP) via Ising
solvers. We show the exactness of a convex copositive reformulation of MBQPs,
allowing the resulting reformulation to inherit the straightforward analysis of
convex optimization. We propose to solve this reformulation with a hybrid
quantum-classical cutting-plane algorithm. Using existing complexity results
for convex cutting-plane algorithms, we deduce that the classical portion of
this hybrid framework is guaranteed to be polynomial time. This suggests that
when applied to NP-hard problems, the complexity of the solution is shifted
onto the subroutine handled by the Ising solver
An Analytic Center Cutting Plane Method to Determine Complete Positivity of a Matrix
We propose an analytic center cutting plane method to determine if a matrix
is completely positive, and return a cut that separates it from the completely
positive cone if not. This was stated as an open (computational) problem by
Berman, D\"ur, and Shaked-Monderer [Electronic Journal of Linear Algebra,
2015]. Our method optimizes over the intersection of a ball and the copositive
cone, where membership is determined by solving a mixed-integer linear program
suggested by Xia, Vera, and Zuluaga [INFORMS Journal on Computing, 2018]. Thus,
our algorithm can, more generally, be used to solve any copositive optimization
problem, provided one knows the radius of a ball containing an optimal
solution. Numerical experiments show that the number of oracle calls (matrix
copositivity checks) for our implementation scales well with the matrix size,
growing roughly like for matrices. The method is
implemented in Julia, and available at
https://github.com/rileybadenbroek/CopositiveAnalyticCenter.jl.Comment: 16 pages, 1 figur
Decomposition Algorithms for Stochastic Programming on a Computational Grid
We describe algorithms for two-stage stochastic linear programming with
recourse and their implementation on a grid computing platform. In particular,
we examine serial and asynchronous versions of the L-shaped method and a
trust-region method. The parallel platform of choice is the dynamic,
heterogeneous, opportunistic platform provided by the Condor system. The
algorithms are of master-worker type (with the workers being used to solve
second-stage problems, and the MW runtime support library (which supports
master-worker computations) is key to the implementation. Computational results
are presented on large sample average approximations of problems from the
literature.Comment: 44 page
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