Search arXiv⌕ Search

SEARCH · Search arXiv

Results for “math.OC”

Search indexed arXiv papers on artificial intelligence, large language models, computer vision and robotics. Read source abstracts and follow links to arXiv.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 19 recordsLinked to original sources
arXiv2020

Chordal-TSSOS: a moment-SOS hierarchy that exploits term sparsity with chordal extension

This work is a follow-up and a complement to arXiv:1912.08899 [math.OC] for solving polynomial optimization problems (POPs). The chordal-TSSOS hierarchy that we propose is a new sparse moment-SOS framework based on term-sparsity and chordal extension. By exploiting term-sparsity of the input polynomials we obtain a two-level hierarchy of semidefinite programming relaxations. The novelty and distinguishing feature of such relaxations is to obtain quasi block-diagonal matrices obtained in an iterative procedure that performs chordal extension of certain adjacency graphs. The graphs are related to the terms arising in the original data and not to the links between variables. Various numerical examples demonstrate the efficiency and the scalability of this new hierarchy for both unconstrained and constrained POPs. The two hierarchies are complementary. While the former TSSOS arXiv:1912.08899 [math.OC] has a theoretical convergence guarantee, the chordal-TSSOS has superior performance but lacks this theoretical guarantee.

math.OC↗
arXiv2024

Input Regularization for Integer Optimal Control in BV with Applications to Control of Poroelastic and Poroviscoelastic Systems

We revisit a class of integer optimal control problems for which a trust-region method has been proposed and analyzed in arXiv:2106.13453v3 [math.OC]. While the algorithm proposed in arXiv:2106.13453v3 [math.OC] successfully solves the class of optimization problems under consideration, its convergence analysis requires restrictive regularity assumptions. There are many examples of integer optimal control problems involving partial differential equations where these regularity assumptions are not satisfied. In this article we provide a way to bypass the restrictive regularity assumptions by introducing an additional partial regularization of the control inputs by means of mollification and proving a $Γ$-convergence-type result when the support parameter of the mollification is driven to zero. We highlight the applicability of this theory in the case of fluid flows through deformable porous media equations that arise in biomechanics. We show that the regularity assumptions are violated in the case of poro-visco-elastic systems, and thus one needs to use the regularization of the control input introduced in this article. Associated numerical results show that while the homotopy can help to find better objective values and points of lower instationarity, the practical performance of the algorithm without the input regularization may be on par with the homotopy.

math.OC↗
arXiv2016

Connection between MP and DPP for Stochastic Recursive Optimal Control Problems: Viscosity Solution Framework in General Case

This paper deals with a stochastic recursive optimal control problem, where the diffusion coefficient depends on the control variable and the control domain is not necessarily convex. We focus on the connection between the general maximum principle and the dynamic programming principle for such control problem without the assumption that the value is smooth enough, the set inclusions among the sub- and super-jets of the value function and the first-order and second-order adjoint processes as well as the generalized Hamiltonian function are established. Moreover, by comparing these results with the classical ones in Yong and Zhou [{\em Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999}], it is natural to obtain the first- and second-order adjoint equations of Hu [{\em Direct method on stochastic maximum principle for optimization with recursive utilities, arXiv:1507.03567v1 [math.OC], 13 Jul. 2015}].

math.OC↗
arXiv2019

Duality of convex relaxations for constrained variational problems

We prove weak duality between two recent convex relaxation methods for bounding the optimal value of a constrained variational problem in which the objective is an integral functional. The first approach, proposed by Valmorbida et al. (IEEE Trans. Automat. Control 61(6):1649--1654, 2016), replaces the variational problem with a convex program over sufficiently smooth functions, subject to pointwise non-negativity constraints. The second approach, discussed by Korda et al. (arXiv:1804.07565v1 [math.OC]), relaxes the variational problem into a convex program over scaled probability measures. We also prove that the duality between these infinite-dimensional convex programs is strong, meaning that their optimal values coincide, when the range and gradients of admissible functions in the variational problem are constrained to bounded sets. For variational problems with polynomial data, the optimal values of each convex relaxation can be approximated by solving weakly dual hierarchies of finite-dimensional semidefinite programs (SDPs). These are strongly dual under standard constraint qualification conditions irrespective of whether strong duality holds at the infinite-dimensional level. Thus, the two relaxation approaches are equivalent for the purposes of computations.

math.OC↗
arXiv2021

Model Predictive Control of a Vehicle using Koopman Operator

This paper continues in the work from arXiv:1903.06103 [math.OC] where a nonlinear vehicle model was approximated in a purely data-driven manner by a linear predictor of higher order, namely the Koopman operator. The vehicle system typically features a lot of nonlinearities such as rigid-body dynamics, coordinate system transformations and most importantly the tire. These nonlinearities are approximated in a predefined subset of the state-space by the linear Koopman operator and used for a linear Model Predictive Control (MPC) design in the high-dimension state space where the nonlinear system dynamics evolve linearly. The result is a nonlinear MPC designed by linear methodologies. It is demonstrated that the Koopman-based controller is able to recover from a very unusual state of the vehicle where all the aforementioned nonlinearities are dominant. The controller is compared with a controller based on a classic local linearization and shortcomings of this approach are discussed.

math.OC↗
arXiv2024

Disciplined Saddle Programming

We consider convex-concave saddle point problems, and more generally convex optimization problems we refer to as $\textit{saddle problems}$, which include the partial supremum or infimum of convex-concave saddle functions. Saddle problems arise in a wide range of applications, including game theory, machine learning, and finance. It is well known that a saddle problem can be reduced to a single convex optimization problem by dualizing either the convex (min) or concave (max) objectives, reducing a min-max problem into a min-min (or max-max) problem. Carrying out this conversion by hand can be tedious and error prone. In this paper we introduce $\textit{disciplined saddle programming}$ (DSP), a domain specific language (DSL) for specifying saddle problems, for which the dualizing trick can be automated. The language and methods are based on recent work by Juditsky and Nemirovski arXiv:2102.01002 [math.OC], who developed the idea of conic-representable saddle point programs, and showed how to carry out the required dualization automatically using conic duality. Juditsky and Nemirovski's conic representation of saddle problems extends Nesterov and Nemirovski's earlier development of conic representable convex problems; DSP can be thought of as extending disciplined convex programming (DCP) to saddle problems. Just as DCP makes it easy for users to formulate and solve complex convex problems, DSP allows users to easily formulate and solve saddle problems. Our method is implemented in an open-source package, also called DSP.

math.OC↗
arXiv2025

Correction to: A Lagrangian dual method for two-stage robust optimization with binary uncertainties

We provide a correction to the sufficient conditions under which closed-form expressions for the optimal Lagrange multiplier are provided in arXiv:2112.13138 [math.OC]. We first present a simple counterexample where the original conditions are insufficient, highlight where the original proof fails, and then provide modified conditions along with a correct proof of their validity. Finally, although the original paper discusses modifications to their method for problems that may not satisfy any sufficient conditions, we substantiate that discussion along two directions. We first show that computing an optimal Lagrange multiplier can still be done in polynomial time. We then provide complete and correct versions of the corresponding Benders and column-and-constraint generation algorithms in which the original method is used. We also discuss the implications of our findings on computational performance.

math.OC↗
arXiv2024

Further extensions on the successive approximation method for hierarchical optimal control problems and its application to learning

In this paper, further extensions of the result of the paper "A successive approximation method in functional spaces for hierarchical optimal control problems and its application to learning, arXiv:2410.20617 [math.OC], 2024" concerning a class of learning problem of point estimations for modeling of high-dimensional nonlinear functions are given. In particular, we present two viable extensions within the nested algorithm of the successive approximation method for the hierarchical optimal control problem, that provide better convergence property and computationally efficiency, which ultimately leading to an optimal parameter estimate. The first extension is mainly concerned with the convergence property of the steps involving how the two agents, i.e., the "leader" and the "follower," update their admissible control strategies, where we introduce augmented Hamiltonians for both agents and we further reformulate the admissible control updating steps as as sub-problems within the nested algorithm of the hierarchical optimal control problem that essentially provide better convergence property. Whereas the second extension is concerned with the computationally efficiency of the steps involving how the agents update their admissible control strategies, where we introduce intermediate state variable for each agent and we further embed the intermediate states within the optimal control problems of the "leader" and the "follower," respectively, that further lend the admissible control updating steps to be fully efficient time-parallelized within the nested algorithm of the hierarchical optimal control problem.

math.OC↗
arXiv2006

Motion camouflage in three dimensions

We formulate and analyze a three-dimensional model of motion camouflage, a stealth strategy observed in nature. A high-gain feedback law for motion camouflage is formulated in which the pursuer and evader trajectories are described using natural Frenet frames (or relatively parallel adapted frames), and the corresponding natural curvatures serve as controls. The biological plausibility of the feedback law is discussed, as is its connection to missile guidance. Simulations illustrating motion camouflage are also presented. This paper builds on recent work on motion camouflage in the planar setting [1]. [1] E.W. Justh and P.S. Krishnaprasad (2005). "Steering laws for motion camouflage," submitted for publication, (see also arXiv:math.OC/0508023v1).

math.OC↗
arXiv2004

Integral functionals on Sobolev spaces having multiple local minima

In this paper, two multiplicity results about local minima of integrals of the calculus of variations are established. The main tool used to prove them is the theory developed in [B. Ricceri, Sublevel sets and global minima of coercive functionals and local minima of their pertubations, math.OC/0402444].

math.OC↗
arXiv2026

A generalized canonical metric for optimization on the indefinite Stiefel manifold

Various tasks in scientific computing can be modeled as an optimization problem on the indefinite Stiefel manifold. We address this using the Riemannian approach, which basically consists of equipping the feasible set with a Riemannian metric, preparing geometric tools such as orthogonal projections, formulae for Riemannian gradient, retraction and then extending an unconstrained optimization algorithm on the Euclidean space to the established manifold. The choice for the metric undoubtedly has a great influence on the method. In the previous work [D.V. Tiep and N.T. Son, A Riemannian gradient descent method for optimization on the indefinite Stiefel manifold, arXiv:2410.22068v2[math.OC]], a tractable metric, which is indeed a family of Riemannian metrics defined by a symmetric positive-definite matrix depending on the contact point, has been used. In general, it requires solving a Lyapunov matrix equation every time when the gradient of the cost function is needed, which might significantly contribute to the computational cost. To address this issue, we propose a new Riemannian metric for the indefinite Stiefel manifold. Furthermore, we construct the associated geometric structure, including a so-called quasi-geodesic and propose a retraction based on this curve. We then numerically verify the performance of the Riemannian gradient descent method associated with the new geometry and compare it with the previous work.

math.OC↗
arXiv2006

An intrinsic metric for power spectral density functions

We present an intrinsic metric that quantifies distances between power spectral density functions. The metric was derived by the author in a recent arXiv-report (math.OC/0607026) as the geodesic distance between spectral density functions with respect to a particular pseudo-Riemannian metric motivated by a quadratic prediction problem. We provide an independent verification of the metric inequality and discuss certain key properties of the induced topology.

math.OC↗
arXiv2022

Bayesian Learning with Wasserstein Barycenters

We introduce and study a novel model-selection strategy for Bayesian learning, based on optimal transport, along with its associated predictive posterior law: the Wasserstein population barycenter of the posterior law over models. We first show how this estimator, termed Bayesian Wasserstein barycenter (BWB), arises naturally in a general, parameter-free Bayesian model-selection framework, when the considered Bayesian risk is the Wasserstein distance. Examples are given, illustrating how the BWB extends some classic parametric and non-parametric selection strategies. Furthermore, we also provide explicit conditions granting the existence and statistical consistency of the BWB, and discuss some of its general and specific properties, providing insights into its advantages compared to usual choices, such as the model average estimator. Finally, we illustrate how this estimator can be computed using the stochastic gradient descent (SGD) algorithm in Wasserstein space introduced in a companion paper arXiv:2201.04232v2 [math.OC], and provide a numerical example for experimental validation of the proposed method.

stat.ML↗
arXiv2026

Invariant manifolds and transversality for a traveling-wave ODE in a controlled reaction-diffusion model

We study the parameter dependence and transversality properties of trajectories of the controlled ODE $$ U_x = P, \qquad P_x = - f(U) - βP + \tilde α(x)U, $$ arising from traveling-wave solutions of a bistable reaction-diffusion equation with control. Optimal profiles minimizing the $L^1$-norm of the control $\tilde α$ are determined by the unstable and stable manifolds $Γ_u$, $Γ_s$, the critical curve $$ P^*(U) = \sqrt{Uf(U)}, $$ and the associated function $$ F(U) = \frac{dP^*}{dU} + \frac{f(U)}{P^*(U)}. $$ We establish generic transversality properties for these objects with respect to perturbations of the speed $β$ and the nonlinear source $f$, together with quantitative estimates on the first- and second-order dependence of the invariant manifolds on these parameters. We also analyze the loss of regularity at the critical speed $β^{**}$, where the unstable manifold approaches the strongly stable manifold of the equilibrium $(U^*,0)$. These results provide the ODE ingredients used in the analysis of the structure and regularity of optimal profiles in paper [S. Bianchini and C. Trifone. Traveling profiles and control cost for a PDE describing the evolution of invasive species. 2026. arXiv: 2606.23979 [math.OC]].

math.AP↗
arXiv2019

Well-posedness and Stability for Interconnection Structures of Port-Hamiltonian Type

We consider networks of infinite-dimensional port-Hamiltonian systems $\mathfrak{S}_i$ on one-dimensional spatial domains. These subsystems of port-Hamiltonian type are interconnected via boundary control and observation and are allowed to be of distinct port-Hamiltonian orders $N_i \in \mathbb{N}$. Wellposedness and stability results for port-Hamiltonian systems of fixed order $N \in \mathbb{N}$ are thereby generalised to networks of such. The abstract theory is applied to some particular model examples.

math.AP↗
arXiv2010

On Verifiable Sufficient Conditions for Sparse Signal Recovery via $\ell_1$ Minimization

We propose novel necessary and sufficient conditions for a sensing matrix to be "$s$-good" - to allow for exact $\ell_1$-recovery of sparse signals with $s$ nonzero entries when no measurement noise is present. Then we express the error bounds for imperfect $\ell_1$-recovery (nonzero measurement noise, nearly $s$-sparse signal, near-optimal solution of the optimization problem yielding the $\ell_1$-recovery) in terms of the characteristics underlying these conditions. Further, we demonstrate (and this is the principal result of the paper) that these characteristics, although difficult to evaluate, lead to verifiable sufficient conditions for exact sparse $\ell_1$-recovery and to efficiently computable upper bounds on those $s$ for which a given sensing matrix is $s$-good. We establish also instructive links between our approach and the basic concepts of the Compressed Sensing theory, like Restricted Isometry or Restricted Eigenvalue properties.

math.OC↗
arXiv2020

Fenchel Duality Theory and A Primal-Dual Algorithm on Riemannian Manifolds

This paper introduces a new notion of a Fenchel conjugate, which generalizes the classical Fenchel conjugation to functions defined on Riemannian manifolds. We investigate its properties, e.g.,~the Fenchel--Young inequality and the characterization of the convex subdifferential using the analogue of the Fenchel--Moreau Theorem. These properties of the Fenchel conjugate are employed to derive a Riemannian primal-dual optimization algorithm, and to prove its convergence for the case of Hadamard manifolds under appropriate assumptions. Numerical results illustrate the performance of the algorithm, which competes with the recently derived Douglas--Rachford algorithm on manifolds of nonpositive curvature. Furthermore, we show numerically that our novel algorithm even converges on manifolds of positive curvature.

math.NA↗