arXiv subjects

Petre Birtea

Publications and source records attributed to Petre Birtea.

At least 19 recordsLinked to original sources

arXiv2026

Optimization with inequality constraints by the embedded gradient vector field method

We develop a geometric framework for constrained optimization problems with inequality constraints through the introduction of quadratic slack variables. This formulation makes it possible to employ the language of Riemannian geometry and to solve the problem via the embedded gradient vector field method. We lift the feasible set to a smooth submanifold of an extended ambient space. The stratified structure of the resulting constraint manifold is analyzed in detail, yielding a natural partition according to which constraints are active. Using the embedded gradient vector field formalism, we derive explicit, determinantal formulas for the Lagrange multiplier functions directly from the geometry of the constraint manifold, recovering and re-framing the classical Karush-Kuhn-Tucker first-order necessary conditions without invoking the classical Lagrange multiplier method. Second-order optimality conditions are obtained by computing the restricted Hessian on each stratum, and a complete sign condition on the Lagrange multipliers is identified as the geometric counterpart of the classical complementary slackness condition. The theory is illustrated on the double ice-cream cone example, where the geometry of the problem determines the nature and number of local minima.

math.OC↗

arXiv2025

An explicit formula for the Laplace-Beltrami operator on the Stiefel manifold

We derive an explicit formula for the Laplace-Beltrami operator on the orthogonal Stiefel manifold, viewed as a constraint submanifold of the Euclidean space of real matrices equipped with the Frobenius metric. Using the general framework of Laplace operators on constraint manifolds, we provide the formula for the Laplace-Beltrami operator in terms of the ambient Euclidean coordinates. The result extends previously known cases, recovering the formulas for the sphere and the special orthogonal group as particular instances.

math.DG↗

arXiv2025

Upper block triangular form for the Laplace-Beltrami operator on the special orthogonal group acquired through a flag of trace polynomials spaces

The Laplacian of a general trace polynomial function defined on the special orthogonal group $SO(N)$ is explicitly computed. An invariant flag of spaces generated by trace polynomials is constructed. The matrix of the Laplace-Beltrami operator on $SO(N)$ for this flag of vector spaces takes an upper block triangular form. As a consequence of this construction, the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on $SO(N)$ can be computed in an iterative manner. For the particular cases of the special orthogonal groups $SO(3)$ and $SO(4)$ the complete list of eigenvalues is obtained and the corresponding irreducible characters of the representation for these groups are expressed as trace polynomials.

math-ph↗

arXiv2025

Generalized double bracket vector fields

We generalize double bracket vector fields, originally defined on semisimple Lie algebras, to Poisson manifolds equipped with a pseudo-Riemannian metric by utilizing a symmetric contravariant 2-tensor field. We extend the normal metric on an adjoint orbit of a compact semisimple Lie algebra to ensure that these vector fields become gradient vector fields on each symplectic leaf. Furthermore, we apply this construction to enhance the equilibria of Hamiltonian systems, specifically addressing the challenge of asymptotically stabilizing points that are already stable, through dissipation terms derived from generalized double bracket vector fields.

math.DG↗

arXiv2023

Laplace-Beltrami operator on the orthogonal group in Cartesian coordinates

Using the embedded gradient vector field method (see P. Birtea, D. Comanescu, Hessian operators on constraint manifolds, J. Nonlinear Science 25, 2015), we present a general formula for the Laplace-Beltrami operator defined on a constraint manifold, written in the ambient coordinates. Regarding the orthogonal group as a constraint submanifold of the Euclidean space of $n\times n$ matrices, we give an explicit formula for the Laplace-Beltrami operator on the orthogonal group using the ambient Euclidean coordinates. We apply this new formula for some relevant functions.

math-ph↗

arXiv2022

Parametric Lie group structures on the probabilistic simplex and generalized Compositional Data

In this paper we build a set of parametric quotient Lie group structures on the probabilistic simplex that can be extended to real vector space structures. In particular, we rediscover the main mathematical objects generally used when treating compositional data as elements associated to the quotient Lie group with respect to the equivalence relation induced by the scale invariance principle. This perspective facilitates the adaptation of the statistical methods used for classical compositional data to data that follows a more general equivalence relation.

math.ST↗

arXiv2021

Constraint optimization and $\mathcal{SU}(N)$ quantum control landscapes

We develop the embedded gradient vector field method, introduced in [8] and [9], for the case of the special unitary group $\mathcal{SU}(N)$ regarded as a constraint submanifold of the unitary group $\mathcal{U}(N)$. The optimization problem associated to the trace fidelity cost function defined on $\mathcal{SU}(N)$ that appears in the context of $\mathcal{SU}(N)$ quantum control landscapes is completely solved using the embedded gradient vector field method. We prove that for $N\geq 5$, the landscape is not $\mathcal{SU}(N)$-trap free, there are always kinematic local extrema that are not global extrema.

quant-ph↗

arXiv2020

Second order optimality on orthogonal Stiefel manifolds

The main tool to study a second order optimality problem is the Hessian operator associated to the cost function that defines the optimization problem. By regarding an orthogonal Stiefel manifold as a constraint manifold embedded in an Euclidean space we obtain a concise matrix formula for the Hessian of a cost function defined on such a manifold. We introduce an explicit local frame on an orthogonal Stiefel manifold in order to compute the components of the Hessian matrix of a cost function. We present some important properties of this frame. As applications we rediscover second order conditions of optimality for the Procrustes and the Penrose regression problems (previously found in the literature). For the Brockett problem we find necessary and sufficient conditions for a critical point to be a local minimum. Since many optimization problems are approached using numerical algorithms, we give an explicit description of the Newton algorithm on orthogonal Stiefel manifolds.

math.NA↗

arXiv2019

Optimization on the symplectic group

We regard the real symplectic group $Sp(2n,\mathbb{R})$ as a constraint submanifold of the $2n\times 2n$ real matrices $\mathcal{M}_{2n}(\mathbb{R})$ endowed with the Euclidean (Frobenius) metric, respectively as a submanifold of the general linear group $Gl(2n,\mathbb{R})$ endowed with the (left) invariant metric. For a cost function that defines an optimization problem on the real symplectic group we give a necessary and sufficient condition for critical points and we apply this condition to the particular case of a least square cost function. In order to characterize the critical points we give a formula for the Hessian of a cost function defined on the real symplectic group, with respect to both considered metrics. For a generalized Brockett cost function we present a necessary condition and a sufficient condition for local minimum. We construct a retraction map that allows us to detail the steepest descent and embedded Newton algorithms for solving an optimization problem on the real symplectic group.

math.OC↗

arXiv2019

Characterization of the critical points for the free energy of a Cosserat problem

Using the embedded gradient vector field method we explicitly compute the list of critical points of the free energy for a Cosserat body model. We also formulate necessary and sufficient conditions for critical points in the abstract case of the special orthogonal group SO(n). Each critical point is then characterized using an explicit formula for the Hessian operator of a cost function defined on the orthogonal group. We also give a positive answer to an open question posed in L. Borisov, A. Fischle, P. Neff, "Optimality of the relaxed polar factors by a characterization of the set of real square roots of real symmetric matrices", ZAMM (2019), namely if all local minima of the optimization problem are global minima. We point out a few examples with physical relevance, in contrast to some theoretical (mathematical) situations that do not hold such a relevance.

math-ph↗

arXiv2018

A New Training Method for Feedforward Neural Networks Based on Geometric Contraction Property of Activation Functions

We propose a new training method for a feedforward neural network having the activation functions with the geometric contraction property. The method consists of constructing a new functional that is less nonlinear in comparison with the classical functional by removing the nonlinearity of the activation function from the output layer. We validate this new method by a series of experiments that show an improved learning speed and better classification error.

cs.NE↗

arXiv2017

Steepest descent algorithm on orthogonal Stiefel manifolds

Considering orthogonal Stiefel manifolds as constraint manifolds, we give an explicit description of a set of local coordinates that also generate a basis for the tangent space in any point of the orthogonal Stiefel manifolds. We show how this construction depends on the choice of a submatrix of full rank. Embedding a gradient vector field on an orthogonal Stiefel manifold in the ambient space, we give explicit necessary and sufficient conditions for a critical point of a cost function defined on such manifolds. We explicitly describe the steepest descent algorithm on the orthogonal Stiefel manifold using the ambient coordinates and not the local coordinates of the manifold. We point out the dependence of the recurrence sequence that defines the algorithm on the choice of a full rank submatrix. We illustrate the algorithm in the case of Brockett cost functions.

math.OC↗

arXiv2016

Dynamics under Geometric Dissipation

We give sufficient conditions for asymptotic stabilization of equilibrium points and periodic orbits of a dynamical system when we add a geometric dissipation of gradient type. We also describe the domain of attraction in the case of asymptotic stability.

math-ph↗

arXiv2016

Newton Algorithm on Constraint Manifolds and the 5-electron Thomson problem

We give a description of numerical Newton algorithm on a constraint manifold using only the ambient coordinates (usually Euclidean coordinates) and the geometry of the constraint manifold. We apply the numerical Newton algorithm on a sphere in order to find the critical configurations of the 5-electron Thomson problem. As a result, we find a new critical configuration of a regular pentagonal type. We also make an analytical study of the critical configurations found previously and determine their nature using Morse-Bott theory. Last section contains an analytical study of critical configurations for Riesz s-energy of 5-electron on a sphere and their bifurcation behavior is pointed out.

math-ph↗

arXiv2015

Hessian Operators on Constraint Manifolds

On a constraint manifold we give an explicit formula for the Hessian matrix of a cost function that involves the Hessian matrix of a prolonged function and the Hessian matrices of the constraint functions. We give an explicit formula for the case of the orthogonal group ${\bf O}(n)$ by using only Euclidean coordinates on $\mathbb{R}^{n^2}$. An optimization problem on ${\bf SO}(3)$ is completely carried out. Its applications to nonlinear stability problems are also analyzed.

math.DG↗

arXiv2014

A note on stability of nongeneric equilibria for an underwater vehicle

We study the Lyapunov stability of a family of nongeneric equilibria with spin for underwater vehicles with noncoincident centers. The nongeneric equilibria belong to singular symplectic leaves that are not characterized as a preimage o a regular value of the Casimir functions. We find an invariant submanifold such that the nongeneric equilibria belong to a preimage of a regular value that involves sub-Casimir functions. We obtain results for nonlinear stability on this invariant submanifold.

math-ph↗

arXiv2014

Double bracket structures on Poisson manifolds

On a Poisson manifold endowed with a Riemannian metric we will construct a vector field that generalizes the double bracket vector field defined on semi-simple Lie algebras. On a regular symplectic leaf we will construct a generalization of the normal metric such that the above vector field restricted to the symplectic leaf is a gradient vector field with respect to this metric.

math.DG↗

arXiv2013

Averaging on Manifolds by Embedding Algorithm

We will propose a new algorithm for finding critical points of cost functions defined on a differential manifold. We will lift the initial cost function to a manifold that can be embedded in a Riemannian manifold (Euclidean space) and will construct a vector field defined on the ambient space whose restriction to the embedded manifold is the gradient vector field of the lifted cost function. The advantage of this method is that it allows us to do computations in Cartesian coordinates instead of using local coordinates and covariant derivatives on the initial manifold. We will exemplify the algorithm in the case of SO(3) averaging problems and will rediscover a few well known results that appear in literature.

math.OC↗