1,507 research outputs found
Interior point methods and simulated annealing for nonsymmetric conic optimization
This thesis explores four methods for convex optimization. The first two are an interior point method and a simulated annealing algorithm that share a theoretical foundation. This connection is due to the interior point method’s use of the so-called entropic barrier, whose derivatives can be approximated through sampling. Here, the sampling will be carried out with a technique known as hit-and-run. By carefully analyzing the properties of hit-and-run sampling, it is shown that both the interior point method and the simulated annealing algorithm can solve a convex optimization problem in the membership oracle setting. The number of oracle calls made by these methods is bounded by a polynomial in the input size. The third method is an analytic center cutting plane method that shows promising performance for copositive optimization. It outperforms the first two methods by a significant margin on the problem of separating a matrix from the completely positive cone. The final method is based on Mosek’s algorithm for nonsymmetric conic optimization. With their scaling matrix, search direction, and neighborhood, we define a method that converges to a near-optimal solution in polynomial time
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 Overview of Integral Quadratic Constraints for Delayed Nonlinear and Parameter-Varying Systems
A general framework is presented for analyzing the stability and performance
of nonlinear and linear parameter varying (LPV) time delayed systems. First,
the input/output behavior of the time delay operator is bounded in the
frequency domain by integral quadratic constraints (IQCs). A constant delay is
a linear, time-invariant system and this leads to a simple, intuitive
interpretation for these frequency domain constraints. This simple
interpretation is used to derive new IQCs for both constant and varying delays.
Second, the performance of nonlinear and LPV delayed systems is bounded using
dissipation inequalities that incorporate IQCs. This step makes use of recent
results that show, under mild technical conditions, that an IQC has an
equivalent representation as a finite-horizon time-domain constraint. Numerical
examples are provided to demonstrate the effectiveness of the method for both
class of systems
Financial Applications of Semidefinite Programming: A Review and Call for Interdisciplinary Research
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
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