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Given a surface M and a fixed conformal class c one defines ƒk.M; c/to be the supremum of the k-th nontrivial Laplacian eigenvalue over all metrics g 2 c of unit volume. It has been observed by Nadirashvili that the metrics achieving ƒk.M; c/are closely related to harmonic maps to spheres. In the present paper, we identify ƒ1.M; c/and ƒ2.M; c/with min-max quantities associated to the energy functional for sphere-valued maps. As an application, we obtain several new eigenvalue bounds, including a sharp isoperimetric inequality for the first two Steklov eigenvalues. This characterization also yields an alternative proof of the existence of maximal metrics realizing ƒ1.M; c/, ƒ2.M; c/, and moreover allows us to obtain a regularity theorem for maximal Radon measures satisfying a natural compactness condition. [ABSTRACT FROM AUTHOR]

In this paper we analyze and solve eigenvalue programs, which consist of the task of minimizing a function subject to constraints on the "eigenvalues" of the decision variable. Here, by making use of the FTvN systems framework introduced by Gowda, we interpret "eigenvalues" in a broad fashion going beyond the usual eigenvalues of matrices. This allows us to shed new light on classical problems such as inverse eigenvalue problems and also leads to new applications. In particular, after analyzing and developing a simple projected gradient algorithm for general eigenvalue programs, we show that eigenvalue programs can be used to express what we call vanishing quadratic constraints. A vanishing quadratic constraint requires that a given system of convex quadratic inequalities be satisfied and at least a certain number of those inequalities must be tight. As a particular case, this includes the problem of finding a point x in the intersection of m ellipsoids in such a way that x is also in the boundary of at least ℓ of the ellipsoids, for some fixed ℓ > 0 . At the end, we also present some numerical experiments. [ABSTRACT FROM AUTHOR]

We investigate how to solve smooth matrix optimization problems with general linear inequality constraints on the eigenvalues of a symmetric matrix. We present solution methods to obtain exact global minima for linear objective functions, i.e., F (X) = ⟨ C , X ⟩ , and perform exact projections onto the eigenvalue constraint set. Two first-order algorithms are developed to obtain first-order stationary points for general non-convex objective functions. Both methods are proven to converge sublinearly when the constraint set is convex. Numerical experiments demonstrate the applicability of both the model and the methods. [ABSTRACT FROM AUTHOR]

A robust ground structure topology optimization framework is presented to handle the uncertainty of load direction and design for the worst case compliance scenario. The deterministic optimization framework is formulated by a min-max compliance objective to first determine the critical load angle corresponding to the worst case compliance and then to design the topology for compliance minimization. The first optimization problem, based on our load definition, is shown to be equivalent to a maximum eigenvalue function, thus causing significant drawbacks in gradient-based optimization approaches in the case of eigenvalue coalescence. Here, we propose a method to treat the non-differentiability of the maximum eigenvalue optimization problem by a smooth maximum regularization function; hence, presenting a framework for optimizing ground structure networks considering an infinite number of load directions. The results achieved demonstrate that the proposed framework provides solutions with low compliance in all possible loading directions leading to robust structural designs. [ABSTRACT FROM AUTHOR]

A two dimensional eigenvalue problem (2DEVP) of a Hermitian matrix pair (A,C) is introduced in this paper. The 2DEVP can be regarded as a linear algebra formulation of the well-known eigenvalue optimization problem of the parameter matrix (A-µC). We first present fundamental properties of the 2DEVP, such as the existence and variational characterizations of 2D-eigenvalues, and then devise a Rayleigh quotient iteration (RQI)-like algorithm, 2DRQI in short, for computing a 2D-eigentriplet of the 2DEVP. The efficacy of the 2DRQI is demonstrated by large scale eigenvalue optimization problems arising from the minmax of Rayleigh quotients and the distance to instability of a stable matrix. [ABSTRACT FROM AUTHOR]

Estimating the absolute pose of a camera given a set of N points and their observations is known as the resectioning or Perspective-n-Point (PnP) problem. It is at the core of most computer vision applications and it can be stated as an instance of three-dimensional registration with point-line distances, making the error quadratic in the unknown pose. The PnP problem, though, is nonconvex due to the constraints associated with the rotation, and iterative algorithms may get trapped into any suboptimal solutions without notice. This work proposes an efficient certification algorithm for central and noncentral cameras that either confirms the optimality of a solution or is inconclusive. We exploit different sets of constraints for the rotation to assess their performance in terms of certification. Two of the formulations lack the Linear Independence Constraint Qualification (LICQ) while one of them has more constraints than variables. This hinders the usage of the "standard" procedure which estimates the Lagrange multipliers in closed-form. To overcome that, we formulate the certification as an eigenvalue optimization and solve it through a line-search method. Our evaluation on synthetic and real data shows that minimal formulations certify most solutions (more than 90\% on real data) whereas redundant formulations are able to certify all of them and even random problem instances. The proposed algorithm runs in microseconds for all these formulations. [ABSTRACT FROM AUTHOR]

Bell inequalities are pillars of quantum physics in that their violations imply that certain properties of quantum physics (e.g., entanglement) cannot be represented by any classical picture of physics. In this article Bell inequalities and their violations are considered through the lens of noncommutative polynomial optimization. Optimality of these violations is certified for a large ma jority of a set of standard Bell inequalities, denoted A2--A89 in the literature. The main techniques used in the paper include the NPA hierarchy, i.e., the noncommutative version of the Lasserre semidefinite programming (SDP) hierarchies based on the Helton--McCullough Positivstellensatz, the Gelfand--Naimark--Segal (GNS) construction with a novel use of the Artin--Wedderburn theory for rounding and projecting, and nonlinear programming (NLP). A new "Newton chip""-like technique for reducing sizes of SDPs arising in the constructed polynomial optimization problems is presented. This technique is based on conditional expectations. Finally, noncommutative Grobner bases are exploited to certify when an optimizer (a solution yielding optimum violation) cannot be extracted from a dual SDP solution. [ABSTRACT FROM AUTHOR]

A discrete variable topology optimization method of internally finned ducts in heat exchangers for efficient thermal performance is proposed. Fully developed convective heat transfer (FDCHT) model, which has been extensively employed and well checked in practical thermal engineering applications, is considered here. Under the uniform wall temperature boundary condition, the energy equation of the FDCHT model can be mathematically formulated as a generalized eigenvalue equation, and the total thermal resistance reflecting the thermal performance can be related to the eigenvalue. The well-known eigenvalue optimization formulation and sensitivity analysis is applied to this optimization problem. Significantly, the physical reality, such as precise Nusselt number, is maintained by using the discrete variable method, i.e., Sequential Approximate Integer Programming. Here, only the 0–1 densities denoted to solid and fluid are involved so that blurry intermediate zones and interpolation schemes are avoided. The practical engineering conditions of fixed pumping power and fixed fluid volume flow rate are discussed under the unified framework, respectively. Several designs of internally finned ducts without any blurry zone are obtained. The optimized design conforms to the three-dimensional precise Conjugate Heat Transfer calculation. Numerical results show that contrary to the design method by predefined heat transfer coefficient, the proposed method can automatically obtain the optimal shape and spacing of fins since the spatially varying effect of convective heat transfer has been achieved. [ABSTRACT FROM AUTHOR]

For a regular polyhedron (or polygon) centered at the origin, the coordinates of the vertices are eigenvectors of the graph Laplacian for the skeleton of that polyhedron (or polygon) associated with the first nontrivial eigenvalue. In this paper, we generalize this relationship. For a given graph, we study the eigenvalue optimization problem of maximizing the first nontrivial eigenvalue of the graph Laplacian over nonnegative edge weights. We show that the spectral realization of the graph using the eigenvectors corresponding to the solution of this problem, under certain assumptions, is a centered, unit-distance graph realization that has maximal total variance. This result gives a new method for generating unit-distance graph realizations and is based on convex duality. A drawback of this method is that the dimension of the realization is given by the multiplicity of the extremal eigenvalue, which is typically unknown prior to solving the eigenvalue optimization problem. Our results are illustrated with a number of examples. [ABSTRACT FROM AUTHOR]

In this work, damping properties of bending viscoelastic thin structures are enhanced by topology optimization. Homogeneous linear viscoelastic plates are optimized and compared when modeled by either the Kirchhoff–Love or Reissner–Mindlin plate theories as well as by the bulk 3D viscoelastic constitutive equations. Mechanical equations are numerically solved by the finite element method and designs are represented by the level-set approach. High performance computing techniques allow to solve the transient viscoelastic problem for very thin 3D meshes, enabling a wider range of applications. The considered isotropic material is characterized by a generalized Maxwell model accounting for the viscoelasticity of both Young modulus and Poisson’s ratio. Numerical results show considerable design differences according to the chosen mechanical model, and highlights a counter-intuitive section shrinking phenomenon discussed at length. The final numerical example extends the problem to an actual shoe sole application, performing its damping optimization in an industrial context. [ABSTRACT FROM AUTHOR]

There are many inevitable uncertain parameters in engineering systems, and the ignorance of these random factors results in a remarkable reduction in the dynamical system's performance for the eigenvalue topology optimization problem. To address the problem, a reliability-based topology optimization (RBTO) model for lightweight design with frequency constraints is established, which aims to account the probabilistic behavior incurred by material properties and geometric dimensions. Then, the single loop approach is used to tackle the RBTO problem. The sensitivities of eigenvalues with respect to the design and random variables are also deduced, in which the simple and multiple eigenfrequencies are simultaneously considered. Moreover, the threshold projection approach is used to eliminate the checkerboard phenomena and grey elements for the eigenvalue topology optimization problem. Three examples are discussed to exhibit the validity of the proposed RBTO methodology. [ABSTRACT FROM AUTHOR]

We optimize a selection of eigenvalues of the Laplace operator with Dirichlet or Neumann boundary conditions by adjusting the shape of the domain on which the eigenvalue problem is considered. Here, a phase-field function is used to represent the shapes over which we minimize. The idea behind this method is to modify the Laplace operator by introducing phase-field dependent coefficients in order to extend the eigenvalue problem on a fixed design domain containing all admissible shapes. The resulting shape and topology optimization problem can then be formulated as an optimal control problem with PDE constraints in which the phase-field function acts as the control. For this optimal control problem, we establish first-order necessary optimality conditions and we rigorously derive its sharp interface limit. Eventually, we present and discuss several numerical simulations for our optimization problem. [ABSTRACT FROM AUTHOR]

In this paper, we study the optimization algorithm to compute the smallest eigenvalue of the Schrödinger operator with volume constraint. A finite element discretization of this problem is established. We provide the error estimate for the numerical solution. The optimal solution can be approximated by a fixed point iteration scheme. Then a monotonic decreasing algorithm is presented to solve the eigenvalue optimization problem. Numerical simulations demonstrate the efficiency of the method.

In this paper we deal with spectral optimization for the Robin Laplacian on a family of planar domains admitting parallel coordinates, namely a fixed‐width strip built over a smooth closed curve and the exterior of a convex set with a smooth boundary. We show that if the curve length is kept fixed, the first eigenvalue referring to the fixed‐width strip is for any value of the Robin parameter maximized by a circular annulus. Furthermore, we prove that the second eigenvalue in the exterior of a convex domain Ω corresponding to a negative Robin parameter does not exceed the analogous quantity for the exterior of a disk whose boundary has a curvature larger than or equal to the maximum of that for ∂Ω$\partial \Omega$. [ABSTRACT FROM AUTHOR]

This article focuses on optimization of polynomials in noncommuting variables, while taking into account sparsity in the input data. A converging hierarchy of semidefinite relaxations for eigenvalue and trace optimization is provided. This hierarchy is a noncommutative analogue of results due to Lasserre (SIAM J Optim 17(3):822–843, 2006) and Waki et al. (SIAM J Optim 17(1):218–242, 2006). The Gelfand–Naimark–Segal construction is applied to extract optimizers if flatness and irreducibility conditions are satisfied. Among the main techniques used are amalgamation results from operator algebra. The theoretical results are utilized to compute lower bounds on minimal eigenvalue of noncommutative polynomials from the literature. [ABSTRACT FROM AUTHOR]

We consider to solve numerically the shape optimization problems of Dirichlet Laplace eigenvalues subject to volume and perimeter constraints. By combining a level set method with the relaxation approach, the algorithm can perform shape and topological changes on a fixed grid. We use the volume expressions of Eulerian derivatives in shape gradient descent algorithms. Finite element methods are used for discretizations. Two and three-dimensional numerical examples are presented to illustrate the effectiveness of the algorithms. [ABSTRACT FROM AUTHOR]

We provide a new hierarchy of semidefinite programming relaxations, called NCTSSOS, to solve large-scale sparse noncommutative polynomial optimization problems. This hierarchy features the exploitation of term sparsity hidden in the input data for eigenvalue and trace optimization problems. NCTSSOS complements the recent work that exploits correlative sparsity for noncommutative optimization problems by Klep et al. (MP, 2021), and is the noncommutative analogue of the TSSOS framework by Wang et al. (SIAMJO 31: 114–141, 2021, SIAMJO 31: 30–58, 2021). We also propose an extension exploiting simultaneously correlative and term sparsity, as done previously in the commutative case (Wang in CS-TSSOS: Correlative and term sparsity for large-scale polynomial optimization, 2020). Under certain conditions, we prove that the optima of the NCTSSOS hierarchy converge to the optimum of the corresponding dense semidefinite programming relaxation. We illustrate the efficiency and scalability of NCTSSOS by solving eigenvalue/trace optimization problems from the literature as well as randomly generated examples involving up to several thousand variables. [ABSTRACT FROM AUTHOR]

In this paper we study optimization problems involving convex nonlinear semidefinite programming (CSDP). Here we convert CSDP into eigenvalue problem by exact penalty function, and apply the U -Lagrangian theory to the function of the largest eigenvalues, with matrix-convex valued mappings. We give the first-and second-order derivatives of U -Lagrangian in the space of decision variables Rm when transversality condition holds. Moreover, an algorithm frame with superlinear convergence is presented. Finally, we give one application: bilinear matrix inequality (BMI) optimization; meanwhile, list their U V decomposition results. [ABSTRACT FROM AUTHOR]

We consider the self-adjoint two-dimensional Schrödinger operator H μ associated with the differential expression - Δ - μ describing a particle exposed to an attractive interaction given by a measure μ supported in a closed curvilinear strip and having fixed transversal one-dimensional profile measure μ ⊥ . This operator has nonempty negative discrete spectrum, and we obtain two optimization results for its lowest eigenvalue. For the first one, we fix μ ⊥ and maximize the lowest eigenvalue with respect to shape of the curvilinear strip; the optimizer in the first problem turns out to be the annulus. We also generalize this result to the situation which involves an additional perturbation of H μ in the form of a positive multiple of the characteristic function of the domain surrounded by the curvilinear strip. Secondly, we fix the shape of the curvilinear strip and minimize the lowest eigenvalue with respect to variation of μ ⊥ , under the constraint that the total profile measure Å > 0 is fixed. The optimizer in this problem is μ ⊥ given by the product of α and the Dirac δ -function supported at an optimal position. [ABSTRACT FROM AUTHOR]

Engineering structures usually operate in some specific frequency bands. An effective way to avoid resonance is to shift the structure's natural frequencies out of these frequency bands. However, in the optimization procedure, which frequency orders will fall into these bands are not known a priori. This makes it difficult to use the existing frequency constraint formulations, which require prescribed orders. For solving this issue, a novel formulation of the frequency band constraint based on a modified Heaviside function is proposed in this paper. The new formulation is continuous and differentiable; thus, the sensitivity of the constraint function can be derived and used in a gradient-based optimization method. Topology optimization for maximizing the structural fundamental frequency while circumventing the natural frequencies located in the working frequency bands is studied. For eliminating the frequently happened numerical problems in the natural frequency topology optimization process, including mode switching, checkerboard phenomena, and gray elements, the "bound formulation" and "robust formulation" are applied. Three numerical examples, including 2D and 3D problems, are solved by the proposed method. Frequency band gaps of the optimized results are obtained by considering the frequency band constraints, which validates the effectiveness of the developed method. [ABSTRACT FROM AUTHOR]

We explicitly classify all S 1 -invariant free boundary minimal annuli and Möbius bands in B n . This classification is obtained from an analysis of the spectrum of the Dirichlet-to-Neumann map for S 1 -invariant metrics on the annulus and Möbius band. First, we determine the supremum of the kth normalized Steklov eigenvalue among all S 1 -invariant metrics on the Möbius band for each k ≥ 1 , and show that it is achieved by the induced metric from a free boundary minimal embedding of the Möbius band into B 4 by kth Steklov eigenfunctions. We then show that the critical metrics of the normalized Steklov eigenvalues on the space of S 1 -invariant metrics on the annulus and Möbius band are the induced metrics on explicit free boundary minimal annuli and Möbius bands in B 3 and B 4 , including some new families of free boundary minimal annuli and Möbius bands in B 4 . Finally, we prove that these are the only S 1 -invariant free boundary minimal annuli and Möbius bands in B n . [ABSTRACT FROM AUTHOR]

In this paper, we mainly consider optimization problems involving the sum of largest eigenvalues of nonlinear symmetric matrices. One of the difficulties with numerical analysis of such problems is that the eigenvalues, regarded as functions of a symmetric matrix, are not differentiable at those points where they coalesce. The U-Lagrangian theory is applied to the function of the sum of the largest eigenvalues, with convex matrix-valued mappings, which doesn't need to be affine. Some of the results generalize the corresponding conclusions for linear mapping. In the approach, we reformulate the first- and second-order derivatives of U-Lagrangian in the space of decision variables RmRm under some mild conditions in terms of VU-space decomposition. We characterize smooth trajectory, along which the function has a second-order expansion. Moreover, an algorithm framework with superlinear convergence is presented. Finally, an application of VU-decomposition derivatives shows that U-Lagrangian possesses proper execution in matrix variable. [ABSTRACT FROM AUTHOR]

A linear time-invariant dissipative Hamiltonian (DH) system x ̇ = (J − R) Q x , with a skew-Hermitian J, a Hermitian positive semidefinite R, and a Hermitian positive definite Q, is always Lyapunov stable and under further weak conditions even asymptotically stable. By exploiting the characterizations from Mehl et al. (SIAM J. Matrix Anal. Appl. 37(4), 1625–1654, 2016), we focus on the estimation of two stability radii for large-scale DH systems, one with respect to non-Hermitian perturbations of R in the form R + BΔCH for given matrices B, C, and another with respect to Hermitian perturbations in the form R + BΔBH,Δ = ΔH. We propose subspace frameworks for both stability radii that converge at a superlinear rate in theory. The one for the non-Hermitian stability radius benefits from the DH structure-preserving model order reduction techniques, whereas for the Hermitian stability radius we derive subspaces yielding a Hermite interpolation property between the full and projected problems. With the proposed frameworks, we are able to estimate the two stability radii accurately and efficiently for large-scale systems which include a finite-element model of an industrial disk brake. [ABSTRACT FROM AUTHOR]

We prove two bounds for the first Robin eigenvalue of the Finsler Laplacian with negative boundary parameter in the planar case. In the constant area problem, we show that the Wulff shape is a maximizer only for values of the boundary parameter, which are close to zero. In the fixed perimeter case, we prove that the Wulff shape is a maximizer of the first eigenvalue for all values of the boundary parameter. [ABSTRACT FROM AUTHOR]

We review some features of topology optimization with a lower bound on the critical load factor, as computed by linearized buckling analysis. The change of the optimized design, the competition between stiffness and stability requirements and the activation of several buckling modes, depending on the value of such lower bound, are studied. We also discuss some specific issues which are of particular interest for this problem, as the use of non-conforming finite elements for the analysis, the use of inconsistent sensitivity that may lead to wrong signs of sensitivities and the replacement of the single eigenvalue constraints with an aggregated measure. We discuss the influence of these practices on the optimization result, giving some recommendations. [ABSTRACT FROM AUTHOR]

In this paper, a shape optimization problem corresponding to the p-Laplacian operator is studied. Given a density function in a rearrangement class generated by a step function, find the density such that the principal eigenvalue is as small as possible. Considering a membrane of known fixed mass and with fixed boundary of prescribed shape consisting of two different materials, our results determine the way to distribute these materials such that the basic frequency of the membrane is minimal. We obtain some qualitative aspects of the optimizer and then we determine nearly optimal sets which are approximations of the minimizer for specific ranges of parameters values. A numerical algorithm is proposed to derive the optimal shape and it is proved that the numerical procedure converges to a local minimizer. Numerical illustrations are provided for different domains to show the efficiency and practical suitability of our approach. [ABSTRACT FROM AUTHOR]

The larger the distance to instability from a matrix is, the more robustly stable the associated autonomous dynamical system is in the presence of uncertainties and typically the less severe transient behavior its solution exhibits. Motivated by these issues, we consider the maximization of the distance to instability of a matrix dependent on several parameters, a nonconvex optimization problem that is likely to be nonsmooth. In the first part we propose a globally convergent algorithm when the matrix is of small size and depends on a few parameters. In the second part we deal with the problems involving large matrices. We tailor a subspace framework that reduces the size of the matrix drastically. The strength of the tailored subspace framework is proven with a global convergence result as the subspaces grow and a superlinear rate-of-convergence result with respect to the subspace dimension. [ABSTRACT FROM AUTHOR]

The article presents an efficient solution method for structural topology optimization aimed at maximizing the fundamental frequency of vibration. Nowadays, this is still a challenging problem mainly because of the high computational cost required by spectral analyses. The proposed method relies on replacing the eigenvalue problem with a frequency response one, which can be tuned and efficiently solved by a multilevel procedure. Connections of the method with multigrid eigenvalue solvers are discussed in details. Several applications demonstrating more than 90% savings of the computational time are presented as well. [ABSTRACT FROM AUTHOR]

This paper is concerned with subspace acceleration techniques for computing the Crawford number, that is, the distance between zero and the numerical range of a matrix A. Our approach is based on an eigenvalue optimization characterization of the Crawford number. We establish local convergence of order 1 +√2 ≈ 2:4 for an existing subspace method applied to such and other eigenvalue optimization problems involving a Hermitian matrix that depends analytically on one parameter. For the particular case of the Crawford number, we show that the relevant part of the objective function is strongly concave. In turn, this enables us to develop a subspace method that only uses three-dimensional subspaces but still achieves global convergence and a local convergence that is at least quadratic. A number of numerical experiments confirm our theoretical results and reveal that the established convergence orders appear to be tight. [ABSTRACT FROM AUTHOR]

Covariate balance is often advocated for objective causal inference since it mimics randomization in observational data. Unlike methods that balance specific moments of covariates, our proposal attains uniform approximate balance for covariate functions in a reproducing-kernel Hilbert space. The corresponding infinite-dimensional optimization problem is shown to have a finite-dimensional representation in terms of an eigenvalue optimization problem. Large-sample results are studied, and numerical examples show that the proposed method achieves better balance with smaller sampling variability than existing methods. [ABSTRACT FROM AUTHOR]

We consider the minimization or maximization of the Jth largest eigenvalue of an analytic and Hermitian matrix-valued function, and build on Mengi, Yildirim, and Kilic [SIAM J. Matrix Anal. Appl., 35, pp. 699{724, 2014]. This work addresses the setting when the matrix-valued function involved is very large. We describe subspace procedures that convert the original problem into a small-scale one by means of orthogonal projections and restrictions to certain subspaces, and that gradually expand these subspaces based on the optimal solutions of small-scale problems. Global convergence and superlinear rate-of-convergence results with respect to the dimensions of the subspaces are presented in the infinite dimensional setting, where the matrix-valued function is replaced by a compact operator depending on parameters. In practice, it succes to solve eigenvalue optimization problems involving matrices with sizes on the scale of tens, instead of the original problem involving matrices with sizes on the scale of thousands. [ABSTRACT FROM AUTHOR]

We consider the problem of distributing two conducting materials in a ball with fixed proportion in order to minimize the first eigenvalue of a Dirichlet operator. It was conjectured that the optimal distribution consists of putting the material with the highest conductivity in a ball around the center. In this paper, we show that the conjecture is false for all dimensions greater than or equal to two.

The search for optimal configurations of pointsets, the most notable examples being the problems of Kepler and Thompson, have an extremely rich history with diverse applications in physics, chemistry, communication theory, and scientific computing. In this paper, we introduce and study a new optimality criteria for pointset configurations. Namely, we consider a certain weighted graph associated with a pointset configuration and seek configurations that minimize certain spectral properties of the adjacency matrix or graph Laplacian defined on this graph, subject to geometric constraints on the pointset configuration. This problem can be motivated by solar cell design and swarming models, and we consider several spectral functions with interesting interpretations such as spectral radius, algebraic connectivity, effective resistance, and condition number. We prove that the regular simplex extremizes several spectral invariants on the sphere. We also consider pointset configurations on flat tori via (i) the analogous problem on lattices and (ii) through a variety of computational experiments. For many of the objectives considered (but not all), the triangular lattice is extremal. [ABSTRACT FROM AUTHOR]

In this work, we propose eigenvalue optimization combined with Lyapunov theory concepts to ensure stability of the Embden-Meyerhof-Parnas pathway, the pentose-phosphate pathway, the phosphotransferase system and fermentation reactions of Escherichia coli. We address the design of a metabolic network for the maximization of different metabolite production rates. The first case study focuses on serine production, based on a model that consists of 18 differential equations corresponding to dynamic mass balances for extracellular glucose and intracellular metabolites, and thirty kinetic rate expressions. A second case study addresses the design problem to maximize ethanol production, based on a dynamic model that involves mass balances for 25 metabolites and 38 kinetic rate equations. The nonlinear optimization problem including stability constraints has been solved with reduced space Successive Quadratic Programming techniques. Numerical results provide useful insights on the stability properties of the studied kinetic models. [ABSTRACT FROM AUTHOR]

In this paper, we study nonlinear optimization problems involving eigenvalues of symmetric matrices. One of the difficulties in solving these problems is that the eigenvalue functions are not differentiable when the multiplicity of the function is not one. We apply the $${\mathcal {U}}$$ -Lagrangian theory to analyze the largest eigenvalue function of a convex matrix-valued mapping which extends the corresponding results for linear mapping in the literature. We also provides the formula of first-and second-order derivatives of the $${\mathcal {U}}$$ -Lagrangian under mild assumptions. These theoretical results provide us new second-order information about the largest eigenvalue function along a suitable smooth manifold, and leads to a new algorithmic framework for analyzing the underlying optimization problem. [ABSTRACT FROM AUTHOR]

Optimization involving eigenvalues arises in a large spectrum of applications in various domains, such as physics, engineering, statistics and finance. In this paper, we consider the arbitrary eigenvalue minimization problems over an afine family of symmetric matrices, which is a special class of eigenvalue function-D.C. function λl. An explicit proximal bundle approach for solving this class of nonsmooth, nonconvex (D.C.) optimization problem is developed. We prove the global convergence of our method, in the sense that every accumulation point of the sequence of iterates generated by the proposed algorithm is stationary. As an application, we use the proposed bundle method to solve some particular eigenvalue problems. Some encouraging preliminary numerical results indicate that our method can solve the test problems eficiently. [ABSTRACT FROM AUTHOR]

We consider an optimization problem for the first Dirichlet eigenvalue of the p -Laplacian on a hypersurface in R 2 n , with n ≥ 2 . If p ≥ 2 n − 1 , then among hypersurfaces in R 2 n which are O ( n ) × O ( n ) -invariant and have one fixed boundary component, there is a surface which maximizes the first Dirichlet eigenvalue of the p -Laplacian. This surface is either Simons’ cone or a C 1 hypersurface, depending on p and n . If n is fixed and p is large, then the maximizing surface is not Simons’ cone. If p = 2 and n ≤ 5 , then Simons’ cone does not maximize the first eigenvalue. [ABSTRACT FROM AUTHOR]

In this paper we study eigenvalue optimization of non-commutative polynomials. That is, we compute the smallest or biggest eigenvalue a non-commutative polynomial can attain. Our algorithm is based on sums of hermittian squares. To test for exactness, the solutions of the dual SDP are investigated. When we consider the eigenvalue lower bounds we can show that attainability of the optimal value on the dual side implies that the eigenvalue bound is attained. We also show how to extract global eigenvalue optimizers with a procedure based on two ingredients: - the first is the solution to the truncated (tracial) moment problem; - the second is the Gelfand-Naimark-Segal (GNS) construction. The implementation of these procedures in our computer algebra system NC-SOStools is presented and several examples pertaining to matrix inequalities are given to illustrate the results.