577 research outputs found
A Newton-bracketing method for a simple conic optimization problem
For the Lagrangian-DNN relaxation of quadratic optimization problems (QOPs),
we propose a Newton-bracketing method to improve the performance of the
bisection-projection method implemented in BBCPOP [to appear in ACM Tran.
Softw., 2019]. The relaxation problem is converted into the problem of finding
the largest zero of a continuously differentiable (except at )
convex function such that if
and otherwise. In theory, the method generates lower
and upper bounds of both converging to . Their convergence is
quadratic if the right derivative of at is positive. Accurate
computation of is necessary for the robustness of the method, but it is
difficult to achieve in practice. As an alternative, we present a
secant-bracketing method. We demonstrate that the method improves the quality
of the lower bounds obtained by BBCPOP and SDPNAL+ for binary QOP instances
from BIQMAC. Moreover, new lower bounds for the unknown optimal values of large
scale QAP instances from QAPLIB are reported.Comment: 19 pages, 2 figure
SDPNAL+: A Matlab software for semidefinite programming with bound constraints (version 1.0)
SDPNAL+ is a {\sc Matlab} software package that implements an augmented
Lagrangian based method to solve large scale semidefinite programming problems
with bound constraints. The implementation was initially based on a majorized
semismooth Newton-CG augmented Lagrangian method, here we designed it within an
inexact symmetric Gauss-Seidel based semi-proximal ADMM/ALM (alternating
direction method of multipliers/augmented Lagrangian method) framework for the
purpose of deriving simpler stopping conditions and closing the gap between the
practical implementation of the algorithm and the theoretical algorithm. The
basic code is written in {\sc Matlab}, but some subroutines in C language are
incorporated via Mex files. We also design a convenient interface for users to
input their SDP models into the solver. Numerous problems arising from
combinatorial optimization and binary integer quadratic programming problems
have been tested to evaluate the performance of the solver. Extensive numerical
experiments conducted in [Yang, Sun, and Toh, Mathematical Programming
Computation, 7 (2015), pp. 331--366] show that the proposed method is quite
efficient and robust, in that it is able to solve 98.9\% of the 745 test
instances of SDP problems arising from various applications to the accuracy of
in the relative KKT residual
An Exceptionally Difficult Binary Quadratic Optimization Problem with Symmetry: a Challenge for The Largest Unsolved QAP Instance Tai256c
Tai256c is the largest unsolved quadratic assignment problem (QAP) instance
in QAPLIB. It is known that QAP tai256c can be converted into a 256 dimensional
binary quadratic optimization problem (BQOP) with a single cardinality
constraint which requires the sum of the binary variables to be 92. As the BQOP
is much simpler than the original QAP, the conversion increases the possibility
to solve the QAP. Solving exactly the BQOP, however, is still very difficult.
Indeed, a 1.48\% gap remains between the best known upper bound (UB) and lower
bound (LB) of the unknown optimal value. This paper shows that the BQOP admits
a nontrivial symmetry, a property that makes the BQOP very hard to solve. The
symmetry induces equivalent subproblems in branch and bound (BB) methods. To
effectively improve the LB, we propose an efficient BB method that incorporates
a doubly nonnegative relaxation, the standard orbit branching and a technique
to prune equivalent subproblems. With this BB method, a new LB with 1.25\% gap
is successfully obtained, and computing an LB with gap is shown to be
still quite difficult.Comment: 19 pages, 7 figures. arXiv admin note: substantial text overlap with
arXiv:2210.1596
Reinforcement Learning Based on Real-Time Iteration NMPC
Reinforcement Learning (RL) has proven a stunning ability to learn optimal
policies from data without any prior knowledge on the process. The main
drawback of RL is that it is typically very difficult to guarantee stability
and safety. On the other hand, Nonlinear Model Predictive Control (NMPC) is an
advanced model-based control technique which does guarantee safety and
stability, but only yields optimality for the nominal model. Therefore, it has
been recently proposed to use NMPC as a function approximator within RL. While
the ability of this approach to yield good performance has been demonstrated,
the main drawback hindering its applicability is related to the computational
burden of NMPC, which has to be solved to full convergence. In practice,
however, computationally efficient algorithms such as the Real-Time Iteration
(RTI) scheme are deployed in order to return an approximate NMPC solution in
very short time. In this paper we bridge this gap by extending the existing
theoretical framework to also cover RL based on RTI NMPC. We demonstrate the
effectiveness of this new RL approach with a nontrivial example modeling a
challenging nonlinear system subject to stochastic perturbations with the
objective of optimizing an economic cost.Comment: accepted for the IFAC World Congress 202
Achieving Constraints in Neural Networks: A Stochastic Augmented Lagrangian Approach
Regularizing Deep Neural Networks (DNNs) is essential for improving
generalizability and preventing overfitting. Fixed penalty methods, though
common, lack adaptability and suffer from hyperparameter sensitivity. In this
paper, we propose a novel approach to DNN regularization by framing the
training process as a constrained optimization problem. Where the data fidelity
term is the minimization objective and the regularization terms serve as
constraints. Then, we employ the Stochastic Augmented Lagrangian (SAL) method
to achieve a more flexible and efficient regularization mechanism. Our approach
extends beyond black-box regularization, demonstrating significant improvements
in white-box models, where weights are often subject to hard constraints to
ensure interpretability. Experimental results on image-based classification on
MNIST, CIFAR10, and CIFAR100 datasets validate the effectiveness of our
approach. SAL consistently achieves higher Accuracy while also achieving better
constraint satisfaction, thus showcasing its potential for optimizing DNNs
under constrained settings
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