SEMI-DEFINITE PROGRAMMING TECHNIQUES FOR STRUCTURED QUADRATIC INVERSE EIGENVALUE PROBLEMS.

We consider semidefinite programming (SDP) formulations of certain truss topology optimization problems, where a lower bound is imposed on the fundamental frequency of vibration of the truss structure. These SDP formulations were introduced in Ohsaki et al. (Comp. Meth. Appl. Mech. Eng. 180:203–217, 1999). We show how one may automatically obtain symmetric designs, by eliminating the ‘redundant’ symmetry in the SDP problem formulation. This has the advantage that the original SDP problem is substantially reduced in size for trusses with large symmetry groups.

A multiscale analysis and computational method based on the Second-Order Two-Scale (SOTS) approach are proposed for the elastic quadratic eigenvalue problems in the periodic composite domain. Two typical quadratic eigenvalue problems with different damping effects are considered, and by the asymptotic expansions of both the eigenfunctions and eigenvalues, the first- and second-order cell functions, the microscale features of this heterogeneous materials are defined successively. Then, the homogenized quadratic eigenvalue problems are derived and the second-order expansions of the eigenfunctions are formed. The eigenvalues are also broadened to the second-order terms by introducing proper auxiliary elastic functions defined in the composite structure, and the nonlinear expressions of the correctors of the eigenvalues are derived. The finite element procedures are established, linearized methods are discussed for solving the quadratic eigenvalue problems and the second-order asymptotic computations are performed. Effectiveness of the asymptotic model is demonstrated by both the qualitative and quantitative comparisons between the computed SOTS approximations and the reference solutions, and the converging behavior of the eigenfunctions are numerically verified. It is also indicated that the second-order correctors are of importance to reconstruct the detailed information of the original eigenfunctions within the micro cells.

An optimal experimental design represents a structured approach to collecting data with the aim of maximizing the information gleaned. Achieving this requires defining an optimality criterion tailored to the specific model under consideration and the purpose of the investigation. However, it is often observed that a design optimized for one criterion may not perform optimally when applied to another. To mitigate this, one strategy involves employing compound designs. These designs balance multiple criteria to create robust experimental plans that are versatile across different applications. In our study, we systematically tackle the challenge of constructing compound approximate optimal experimental designs using Semidefinite Programming. We focus on discretized design spaces, with the objective function being the geometric or the arithmetic mean of design efficiencies relative to individual criteria. We address two combinations of two criteria: concave-concave (illustrated by DE–optimality) and convex-concave (such as DA–optimality). To handle the latter, we reformulate the problem as a bilevel problem. Here, the outer problem is solved using Surrogate Based Optimization, while the inner problem is addressed with a Semidefinite Programming solver. We demonstrate our formulations using both linear and nonlinear models (for the response) of the Beta class, previously linearized to facilitate analysis and comparison.

We present a unified framework for analyzing the convergence of distributed optimization algorithms by formulating a semidefinite program (SDP) which can be efficiently solved to bound the linear rate of convergence. Two different SDP formulations are considered. First, we formulate an SDP that depends explicitly on the gossip matrix of the network graph. This result provides bounds that depend explicitly on the graph topology, but the SDP dimension scales with the size of the graph. Second, we formulate an SDP that depends implicitly on the gossip matrix via its spectral gap. This result provides coarser bounds, but yields a small SDP that is independent of graph size. Our approach improves upon existing bounds for the algorithms we analyzed, and numerical simulations reveal that our bounds are likely tight. The efficient and automated nature of our analysis makes it a powerful tool for algorithm selection and tuning, and for the discovery of new algorithms as well.

A polynomial optimization problem (POP) consists of minimizing a multivariate real polynomial on a semi-algebraic set \(K\) described by polynomial inequalities and equations. In its full generality it is a non-convex, multi-extremal, difficult global optimization problem. More than an decade ago, J. B. Lasserre proposed to solve POPs by a hierarchy of convex semidefinite programming (SDP) relaxations of increasing size. Each problem in the hierarchy has a primal SDP formulation (a relaxation of a moment problem) and a dual SDP formulation (a sum-of-squares representation of a polynomial Lagrangian of the POP). In this note, we show that there is no duality gap between each primal and dual SDP problem in Lasserre’s hierarchy, provided one of the constraints in the description of set \(K\) is a ball constraint. Our proof uses elementary results on SDP duality, and it does not assume that \(K\) has a strictly feasible point.

Numerical solution of a quadratic eigenvalue problem*1

SummaryHeavily damped quadratic eigenvalue problem (QEP) is a special type of QEPs. It has a large gap between small and large eigenvalues in absolute value. One common way for solving QEP is to linearize the original problem via linearizations. Previous work on the accuracy of eigenpairs of not heavily damped QEP focuses on analyzing the backward error of eigenpairs relative to linearizations. The objective of this paper is to explain why different linearizations lead to different errors when computing small and large eigenpairs. To obtain this goal, we bound the backward error of eigenpairs relative to the linearization methods. Using these bounds, we build upper bounds of growth factors for the backward error. We present results of numerical experiments that support the predictions of the proposed methods.

Model updating for quadratic eigenvalue problems (QEPs) is pro- posed by Friswell, Inman and Pilkey (1998), to incorporate the measured model data into the model which only produces the mass and stifiness matrices, that closely match the experimental model data. In this paper, we consider a numer- ical model for updating QEPs which produces not only the mass and stifiness matrices, but also the damping matrix will be updated. To this end the com- plete set of eigenpairs will be employed.

To solve quadratic eigenvalue problems (QEPs), especially the gyroscopic systems, two methods are proposed: an iterative direct detection method (DDM) of the complex eigenvalues of the original QEP, and a split-linearization method (SLM) for finding the solvent matrix, which results to a standard linear eigenvalue problem (LEP) being solved to compute all eigenvalues by the symmetry extension. Reducing the dimension to one-half, the LEP is recast in a simpler QEP involving the square of the solvent. We set up two new merit functions which are minimized to detect the complex eigenvalues from the original QEP and a simpler QEP. For each eigen-parameter the merit function consists of the Euclidean norm of each derived eigen-equation, whose vector variable is solved from a derived inhomogeneous linear system. Then, the golden section search algorithm is employed to minimize the merit functions and locate the complex eigenvalue as a local minimal point. The results are compared with that computed by the cyclic-reduction-based solvent (CRS) method.

We propose a novel formulation of minimal case solutions for determining the relative pose of perspective and generalized cameras given a partially known rotation, namely, a known axis of rotation. An axis of rotation may be easily obtained by detecting vertical vanishing points with computer vision techniques, or with the aid of sensor measurements from a smart phone. Given a known axis of rotation, our algorithms solve for the angle of rotation around the known axis along with the unknown translation. We formulate these relative pose problems as Quadratic Eigen value Problems which are very simple to construct. We run several experiments on synthetic and real data to compare our methods to the current state-of-the-art algorithms. Our methods provide several advantages over alternatives methods, including efficiency and accuracy, particularly in the presence of image and sensor noise as is often the case for mobile devices.

This paper is an attempt to exploit the opportunity of Semi Definite Programming (SDP), which is an area of convex and conic optimization. Indeed, Numerous NP-hard problems can be solveby using this approach. Hence, we intend to investigate the strength of SDP to model and provide tight relaxations of combinatorial and quadratic problems in order to present a new polynomial time algorithm for solving this robust model. This algorithm was firstly use to solve the nonlinear programs, the reason for which we search to extend it to the SDP programs. Actually, this algorithm designs the combination of two penalization methods. The first one is a primal-dual interior point (PDIM) method while the second one is a primal-dual exterior point (PDEM) method. Unlike the first method, which converges globally, the second one, also called the primal dual nonlinear rescaling method, has local super linear/quadratic convergence. Therefore, it seems appropriate to use a mixed algorithm based on the interior-exterior point method (IEPM). In fact, this resolution starts from the interior method, and at a certain level of execution, it proceeds to exterior method. Hence, a convergence evaluation function is use to know the level of permutation. Through evaluation, it has been approve that our approach is use to solve some instances of max-cut problem. This problem is a central graph theory model that occurs in many real problems and it is one of many NP-hard problems, which has attracted many researchers over the years. Then, we have used a semi definite programming solver SDPA (Semi Definite Programming Algorithm) that is modify to include the exterior point method subroutine. From the computational performance, we conclude that as the problem size increases, interior-exterior point algorithm gets relatively faster. The numerical results obtained are promising.

For a given set of input-output pairs of quantum states or observables, we ask the question whether there exists a physically implementable transformation that maps each of the inputs to the corresponding output. The physical maps on quantum states are trace-preserving completely positive maps, but we also consider variants of these requirements. We generalize the definition of complete positivity to linear maps defined on arbitrary subspaces, then formulate this notion as a semidefinite program, and relate it by duality to approximative extensions of this map. This gives a characterization of the maps which can be approximated arbitrarily well as the restriction of a map that is completely positive on the whole algebra, also yielding the familiar extension theorems on operator spaces. For quantum channel extensions and extensions by probabilistic operations we obtain semidefinite characterizations, and we also elucidate the special case of Abelian inputs or outputs. Finally, revisiting a theorem by Alberti and Uhlmann, we provide simpler and more widely applicable conditions for certain extension problems on qubits, and by using a semidefinite programming formulation we exhibit counterexamples to seemingly reasonable but false generalizations of the Alberti-Uhlmann theorem.

A gridless method for direction finding with sparse arrays in nonuniform noise

We consider quadratic eigenvalue problems with large and sparse matrices depending on a parameter. Problems of this type occur, for example, in the stability analysis of spatially discretized and p...

In several applications in science and engineering, different types of matrix problems emerge from the discretization of partial differential equations. This thesis is devoted to the development of new algorithms to solve this kind of problems. In particular, when the matrices involved are sparse and non-symmetric. The new algorithms are based on the Induced Dimension Reduction method [IDR(s)]. IDR(s) is a Krylov subspace method originally proposed in 2008 to solve systems of linear equations. IDR(s) has received considerable attention due to its stable and fast convergence. It is, therefore, natural to ask if it is possible to extend IDR(s) to solve other matrix problems, and if so, to compare those extensions with other well-established methods. This work aims to answer these questions. The main matrix problems considered in this dissertation are: the standard eigenvalue problem, the quadratic eigenvalue problem, the solution of systems of linear equations, the solution of sequences of systems of linear equations, and linear matrix equations. We focus on examples that arise from the discretization of partial differential equations.