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This article presents a unified framework for constrained optimization solved by using damped dynamical systems on the Stiefel manifold, combining variational principles, projected-gradient methodologies, and asymptotic stability theory. For smooth objective functions defined on the Stiefel manifold, first-order optimality conditions are derived using both intrinsic tangent-space projections and classical Lagrange multiplier formulations, which naturally lead to second-order damped dynamical systems whose equilibrium points coincide with the Karush–Kuhn–Tucker solutions of the constrained optimization problem. Two complementary formulations are studied in detail: a Lagrange-based approach in which constraint satisfaction is enforced through dynamically evolving multipliers, and a projected-gradient formulation in which the dynamics evolve intrinsically on the tangent bundle of the Stiefel manifold. It is shown that both formulations admit identical stationary solutions, and explicit analytical relationships between the Lagrangian and projected dynamics are established. The proposed framework is applied to two canonical problems, namely the linear eigenvalue problem for computing invariant subspaces associated with the smallest eigenvalues of a symmetric positive definite matrix and the orthogonal Procrustes problem formulated in the Frobenius norm, for which explicit expressions for the gradients, multiplier dynamics, and reduced systems are derived. A rigorous asymptotic stability analysis is carried out by linearizing the resulting first-order systems and characterizing the spectra of the associated reduced Jacobian operators acting on the tangent space of the constraint manifold, leading to sufficient conditions for asymptotic stability that clarify the role of damping parameters in guaranteeing convergence. Numerical implementations are developed by discretizing the proposed second-order dynamical systems using a symplectic Euler scheme that preserves the qualitative stability properties of the continuous-time models, resulting in algorithms that rely on standard linear algebra operations, including Sylvester equation solvers and thin singular value decompositions, and that enable a direct comparison between intrinsic and extrinsic approaches to optimization on the Stiefel manifold.

This paper examines optimal portfolio selection using quantile-based risk measures such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). We address the case of a singular covariance matrix of asset returns, which may arise due to potential multicollinearity and strong correlations. This leads to an optimization problem with infinitely many solutions. An analytical form for a general solution is derived, along with a unique solution that minimizes the -norm. We show that the general solution reduces to the standard optimal portfolio for VaR and CVaR when the covariance matrix is non-singular. We also provide a brief discussion of the efficient frontier in this context. Finally, we present a real-data example based on the weekly log returns of assets included in the S&P 500 index.

The basis of our approach is a simple second-order damped oscillating dynamical system (a damped mass-spring system with some externalforce). By changing the damping and the force the solution will fast approach a stationary solution with minimal energy.

For an optimization problem with constraints we generalize the idea such that the system converges to a stationary solution which is a local minima, most often aiming for the global minima. By using an additional damped system for the constraints, we obtain explicit expressions for the Lagrange parameters. 

The dynamical system is solved by a symplectic algorithm, that is normally especially tailored for conservative systems.

Our method is easy to implement, either with coding or with software for dynamical systems and can be applied to a large number of problems.

Numerical experiments show good convergence properties in general, that can be tuned by parameters for the damping and force of the oscillator. In particular, the dynamical constraints make the iterations less sensitive to initial conditions, which do not have to fulfill the constraints.

Calculations of eigenvalue problems and ground states of non-linear Schrödinger equations (NLSE) will be presented. We have worked on multi-component nonlinear Schrödinger equations in arbitrary dimensions. Following the experimental developments of quantum droplets of ultra-cold atoms, we are now modeling other forms of nonlinearities. This opens for future research in non-convex optimization, and calculations of higher eigenvalues to non-linear equations.

The unmanageable amount of encoded information in a many-body particles system makes calculations of its dynamic a challenge. Because of the many degrees of correlation between the particle states, the complexity of the many-body state increases exponentially with the number of particles and their available states. A method that can manage this challenge of complexity is the Gaussian Phase-Space Representation (GPSR) [1]. In GPSR, the wave-function is mapped to a density probability of one-particle density matrices, and the time-dependent Schrödingerequation to a Fokker-Planck equation (FPE). The FPE has to be solved with stochastic differential equations but unfortunately, using stochastic processes induces a maximum simulation time, because some trajectories will end up diverging, which nullifies the validity of averages and prevents us from recovering quantum observables. However, there is a freedom in the decomposition of the diffusion matrix D into the noise matrix B, with D = BBT, and we can extend the maximum simulation time by choosing a ‘better’ noise matrix [2, 3]. We present here the results of GPSR with a noise-matrix computed with a random-SVD [4], which doubles the maximum simulation time while keeping the computational cost reasonable. Understanding why this decomposition works so well might help us finding even better decompositions. 

References

[1] Corney, J. F. and Drummond, P. D. Gaussian operator bases for correlated fermions. Journal of Physics A: Mathematical and General, Volume 39, 269 (2005).

[2] Ögren, M., Kheruntsyan, K. V. and Corney, Joel F. Stochastic simulations of fermionic dynamics with phase-space representations. Computer Physics Communications, Volume 182, 1999 (2011).

[3] Rousse, F., Fasi, M., Dmytryshyn, A., Gulliksson, M. and Ögren, M. Simulations of quantum dynamics with fermionic phase-space representations using numerical matrix factorizations as stochastic gauges. Journal of Physics A: Mathematical and Theoretical, Volume 57, 015303, (2024).

[4] Halko, N., Martinsson, P. G. and Tropp, J. A. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM review,Volume 53, 217, (2011).

This paper examines optimal portfolio selection using quantile-based risk measures such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). We address the case of a singular covariance matrix of asset returns, which leads to an optimization problem with infinitely many solutions. An analytical form for a general solution is derived, along with a unique solution that minimizes the L2-norm. We also show that the general solution reduces to the standard optimal portfolio for VaR and CVaR when the covariance matrix is non-singular.

In this paper, we consider optimal portfolio selection when the covariance matrix of the asset returns is rank-deficient. For this case, the original Markowitz' problem does not have a unique solution. The possible solutions belong to either two subspaces namely the range- or nullspace of the covariance matrix. The former case has been treated elsewhere but not the latter. We derive an analytical unique solution, assuming the solution is in the null space, that is risk-free and has minimum norm. Furthermore, we analyse the iterative method which is called the discrete functional particle method in the rank-deficient case. It is shown that the method is convergent giving a risk-free solution and we derive the initial condition that gives the smallest possible weights in the norm. Finally, simulation results on artificial problems as well as real-world applications verify that the method is both efficient and stable.

The Gaussian phase-space representation can be used to implement quantum dynamics for fermionic particles numerically. To improve numerical results, we explore the use of dynamical diffusion gauges in such implementations. This is achieved by benchmarking quantum dynamics of few-body systems against independent exact solutions. A diffusion gauge is implemented here as a so-called noise-matrix, which satisfies a matrix equation defined by the corresponding Fokker-Planck equation of the phase-space representation. For the physical systems with fermionic particles considered here, the numerical evaluation of the new diffusion gauges allows us to double the practical simulation time, compared with hitherto known analytic noise-matrices. This development may have far reaching consequences for future quantum dynamical simulations of many-body systems. 

Recent years have seen a renewal of interest in multi-term linear matrix equations, as these have come to play a role in a number of important applications. Here, we consider the solution of such equations by means of the dynamical functional particle method, an iterative technique that relies on the numerical integration of a damped second order dy-namical system. We develop a new algorithm for the solution of a large class of these equations, a class that includes, among others, all linear matrix equations with Hermi-tian positive definite or negative definite coefficients. In numerical experiments, our MAT -LAB implementation outperforms existing methods for the solution of multi-term Sylvester equations. For the Sylvester equation AX + XB = C, in particular, it can be faster and more accurate than the built-in implementation of the Bartels-Stewart algorithm, when A and B are well conditioned and have very different size.

We present new approaches for solving constrained multicomponent nonlinear Schrödinger equations in arbitrary dimensions. The idea is to introduce an artificial time and solve an extended damped second order dynamic system whose stationary solution is the solution to the time-independent nonlinear Schrödinger equation. Constraints are often considered by projection onto the constraint set, here we include them explicitly into the dynamical system. We show the applicability and efficiency of the methods on examples of relevance in modern physics applications.

In this paper, we consider optimal portfolio selection when the covariance matrix of the asset returns is rank-deficient. For this case, the original Markowitz’ problem does not have a unique solution. The possible solutions belong to either two subspaces namely the range- or nullspace of the covariance matrix. The former case has been treated elsewhere but not the latter. We derive an analytical unique solution, assuming the solution is in the null space, that is risk-free and has minimum norm. Furthermore, we analyse the iterative method which is called the discrete functional particle method in the rank-deficient case. It is shown that the method is convergent giving a risk-free solution and we derive the initial condition that gives the smallest possible weights in the norm. Finally, simulation results on artificial problems as well as real-world applications verify that the method is both efficient and stable.