Constrained Minimization Problems (Method of Lagrange Multipliers)

A direct least-squares formulation of the stiffness matrix adjustment method known as the KMA method (Kabe, A. M., Stiffness Matrix Adjustment Using Mode Data, AIAA Journal, Vol. 23, No. 9, 1985, pp. 1431-1436) is presented. The KMA method belongs to a class of procedures that refine dynamic models using test-measured modes and structural connectivity. Viewed as a constrained minimization problem, most of these methods have applied Lagrange multiplier techniques in their development. It is shown that these two approaches result in linear systems of equations that are algebraically equivalent and therefore have identical solutions. By virtue of this equivalence, the direct least-squares formulation of the KMA method is shown to be equivalent to its original Lagrange multiplier formulation. The direct least-squares versions of other model refinement methods that preserve structural connectivity are also discussed. In this respect, this approach provides a rigorous framework for unifying these optimal update procedures and indicates how their solutions should compare. Numerical results are presented that illustrate the equivalence between the two formulations and also the conditions under which the various methods yield identical solutions. The results also indicate that because of improved numerical conditioning, the direct least-squares approach yields more accurate computational solutions and generally requires less computer storage than methods that were developed using Lagrange multipliers.

The parallel solution of constrained minimization problems requires special care to be taken with respect to the information transfer between the different subproblems. Here, we present a nonlinear decomposition approach which employs an additional nonlinear correction step along the processor interfaces. Our approach is generic in the sense that it can be applied to a wide class of minimization problems with strongly local nonlinearities, including even nonsmooth minimization problems. We also describe the implementation of our nonlinear decomposition method in the object oriented library ObsLib $$++$$++. The flexibility of our approach and its implementation is presented along different problem classes as obstacle problems, frictional contact problems and biomechanical applications. For the same examples, number of iterations, computation time, and parallelization speedup are measured, and the results demonstrate that the implementation scales reasonably well up to 4096 processors.

We study the minimization problem of a non-convex sparsity promoting penalty function, the transformed $$l_1$$ (TL1), and its application in compressed sensing (CS). The TL1 penalty interpolates $$l_0$$ and $$l_1$$ norms through a nonnegative parameter $$a \in (0,+\infty )$$ , similar to $$l_p$$ with $$p \in (0,1]$$ , and is known to satisfy unbiasedness, sparsity and Lipschitz continuity properties. We first consider the constrained minimization problem, and discuss the exact recovery of $$l_0$$ norm minimal solution based on the null space property (NSP). We then prove the stable recovery of $$l_0$$ norm minimal solution if the sensing matrix A satisfies a restricted isometry property (RIP). We formulated a normalized problem to overcome the lack of scaling property of the TL1 penalty function. For a general sensing matrix A, we show that the support set of a local minimizer corresponds to linearly independent columns of A. Next, we present difference of convex algorithms for TL1 (DCATL1) in computing TL1-regularized constrained and unconstrained problems in CS. The DCATL1 algorithm involves outer and inner loops of iterations, one time matrix inversion, repeated shrinkage operations and matrix-vector multiplications. The inner loop concerns an $$l_1$$ minimization problem on which we employ the Alternating Direction Method of Multipliers. For the unconstrained problem, we prove convergence of DCATL1 to a stationary point satisfying the first order optimality condition. In numerical experiments, we identify the optimal value $$a=1$$ , and compare DCATL1 with other CS algorithms on two classes of sensing matrices: Gaussian random matrices and over-sampled discrete cosine transform matrices (DCT). Among existing algorithms, the iterated reweighted least squares method based on $$l_{1/2}$$ norm is the best in sparse recovery for Gaussian matrices, and the DCA algorithm based on $$l_1$$ minus $$l_2$$ penalty is the best for over-sampled DCT matrices. We find that for both classes of sensing matrices, the performance of DCATL1 algorithm (initiated with $$l_1$$ minimization) always ranks near the top (if not the top), and is the most robust choice insensitive to the conditioning of the sensing matrix A. DCATL1 is also competitive in comparison with DCA on other non-convex penalty functions commonly used in statistics with two hyperparameters.

Quadratic constrained maximization appears as the central problem in many filter design methods and signal processing applications, whenever the processing or filtering entails the maximization of a signal to noise ratio. Furthermore, self reference systems for MSE filters, beamformers and equalizers which manage either time or frequency diversity of the desired signal, give rise to the same mathematical problem. This work reports an adaptive algorithm based on two coupled LMS-like algorithms. It is shown how the Lagrange multiplier, involved in constrained minimization problems, plays the role of gain control in forming the error signal for the filter updates. This gain is power controlled by the output signals keeping the proper equilibrium between the outputs of the two LMS loops.

Computer analysis of structures has traditionally been carried out using the displacement method combined with an incremental iterative scheme for nonlinear problems. In this paper, a Lagrangian approach is developed, which is a mixed method, where besides displacements, the stress resultants and other variables of state are primary unknowns. The method can potentially be used for the analysis of collapse of structures subjected to severe vibrations resulting from shocks or dynamic loads. The evolution of the structural state in time is provided a weak formulation using Hamilton's principle. It is shown that a certain class of structures, known as reciprocal structures, has a mixed Lagrangian formulation in terms of displacements and internal forces. The form of the Lagrangian is invariant under finite displacements and can be used in geometric nonlinear analysis. For numerical solution, a discrete variational integrator is derived starting from the weak formulation. This integrator inherits the energy and momentum conservation characteristics for conservative systems and the contractivity of dissipative systems. The integration of each step is a constrained minimization problem and it is solved using an augmented Lagrangian algorithm. In contrast to the traditional displacement-based method, the Lagrangian method provides a generalized formulation which clearly separates the modeling of components from the numerical solution. Phenomenological models of components, essential to simulate collapse, can be incorporated without having to implement model-specific incremental state determination algorithms. The state variables are determined at the global level by the optimization method.

The main object of our consideration in this chapter is the following parameterized minimization problem $$\left ,$$ where \(I_{{\varepsilon}}:(\Xi_{{\varepsilon}}\subseteq \mathbb{X}_{{\varepsilon}})\rightarrow\overline{\mathbb{R}}\) is an objective functional, \(\Xi_{{\varepsilon}}\subseteq \mathbb{X}_{{\varepsilon}}\) is a set of admissible solutions, and \(\mathbb{X}_{{\varepsilon}}\) is some Banach space.

A new sharp high-order interface tracking method for multi-material flow problems on unstructured meshes is presented. This marker re-distancing (MRD) method is designed to work accurately and robustly on unstructured, generally highly distorted meshes, necessitated by applications within ALE-based hydrocodes. The method is a hybrid of a Lagrangian marker tracking method and a novel discontinuous Galerkin (DG) projection based level set re-distancing algorithm. The re-distancing method is formulated as a constrained minimization problem and is shown to obtain arbitrary orders of convergence for smooth interfaces. High-order (>2nd) re-distancing on irregular meshes is a must for applications were the interfacial curvature is important for the underlying physics, such as surface tension, wetting, and detonation shock dynamics. Since no PDE is solved for re-distancing, the method does not have a stability time step restriction, which is particularly useful in combination with AMR, used here to efficiently resolve fine interface features. In addition, the method can robustly handle discontinuities in the distance function without explicit utilization of solution limiters. Results will be shown for a number of different interface geometries, which will demonstrate the method’s capability of obtaining high-fidelity results on arbitrary meshes.

We introduce a new kind of regularity in the Dirichlet problem for the heat equation, called mild regularity. The mildly regular boundary points include all types of regular points previously considered. Just as regular boundary points have barriers associated with them, the mildly regular boundary points have mild barriers, which are just as important. We consider whether the mild regularity of a boundary point for an open set is inherited by an open subset or superset. We detail the connection of mild regularity with the Green function. We consider, in particular, subdomains of an arbitrary open set that satisfy a certain connectivity condition which takes account of the special nature of the temporal variable, called zones of influence. Our main theorem states that, for any zone of influence, the set of boundary points that are not mildly regular is caloric measure null.

studied by Kazhdan and Lusztig in [KL]. For x = gG(O) ∈ X the G(O)-orbit (for the adjoint action) of Ad(g−1)(u) in g(O) depends only on x, and its image under g(O) g(C) is a well-defined G(C)-orbit in g(C). We say that x ∈Xu is regular if the associated orbit is regular in g(C). (Recall that an element of g(C) is regular if the nilpotent part of its Jordan decomposition is a principal nilpotent element in the centralizer of the semisimple part of its Jordan decomposition.) We write X reg for the (Zariski open) subset of regular elements in X. From now on we assume that u is regular semisimple with centralizer T, a maximal torus in G over F. Assume further that u is integral, by which we mean that X is nonempty. Kazhdan and Lusztig [KL] show that X is then a locally finite union of projective algebraic varieties, and in [KL, Sec. 4, Cor. 1] they show that the open subset X reg of X u is nonempty (and hence dense in at least one irreducible component of X). The action of T(F ) on X clearly preserves the subsets X andX reg. Bezrukavnikov [B] proved thatX u reg forms a single orbit under T(F ). (Actually Kazhdan–Lusztig and Bezrukavnikov consider only topologically nilpotent elements u, but the general case can be reduced to their special case by using the topological Jordan decomposition of u.) The goal of this paper is to characterize regular elements in X (for integral regular semisimple u as just described). When T is elliptic (in other words, F anisotropic modulo the center of G), the characterization gives no new information. At the other extreme, in the split case, the characterization gives a clear picture of what it means for a point in X to be regular. We will now state our characterization in the split case, leaving the more technical general statement to the next section (see Theorem 1). Fix a split maximal torus

YAM2: Yet another library for the [formula omitted] variables using sequential quadratic programming

Strong electronic correlations are at the heart of many interesting phenomena. For the theoretical description of these materials, a proper treatment of the local atomic physics is required. We propose a novel approach combining functionals of the electron density and functionals of the one-particle reduced density matrix to improve this description. The proposed method has a solid foundation in reduced density-matrix functional theory and no double-counting problem arises. It employs a decomposition of the electron-electron interaction in real space. The interaction close to the correlated orbitals, for example, the partially filled 3d-orbitals of transition-metal ions, is described with a density-matrix functional and otherwise with a local or semi-local density functional. We propose to evaluate the density-matrix functional from Levy’s constrained search problem, i.e., via a constrained minimization over an ensemble of many-particle wave functions. In contrast to approximate parametrized density-matrix functionals, this evaluation allows us to systematically improve the functional towards the exact result. In situations where the one-particle basis is too large to evaluate the density-matrix functional from Levy’s constrained minimization problem, we apply a series of approximations that each can be converged to the exact result: the first approximation step is the local approximation of the density-matrix functional proposed by Blöchl, Walther and Pruschke. For the density-matrix functionals within the local interactions, we propose the adaptive cluster approximation (ACA) that systematically truncates non-interacting one-particle states and drastically reduces the computational effort. The resulting density-matrix functional for a local interaction and a small number of non-interacting one-particle states is then evaluated with the constrained minimization problem. We explore different parametrizations of many-particle wave functions in this constrained minimization problem. A parametrization based on a configuration-interaction-like ansatz is shown to converge rapidly if suitable selection criterion for the Slater determinants is chosen. An impurity-bath-separation ansatz is shown to be suitable for single-impurity Anderson models. It is shown that the constrained minimization can be solved for matrix product states with a DMRG-like iterative minimization. Furthermore, we show that Gutzwiller-Jastrow-correlated wave functions can be used with a quantum Monte Carlo procedure as many-particle wave functions. Finally, we formulate an algorithm for the evaluation of the density-matrix functional on near-term quantum computers. Results from the execution of the algorithm on an existing quantum computer with transmon qubits are presented. The proposed approach combining density functionals and density-matrix functionals is implemented in the CP-PAW code based on the projector augmented-wave formalism. We present results for the dissociation curve of the hydrogen molecule as the prototypical case of strong static correlation and show that static correlation is well described with the new approach. We show results for the nonmagnetic state of the transition-metaoxide NiO that is described qualitatively wrong with the DFT+U method. The proposed method properly describes the nonmagnetic state of NiO as an insulator and predicts a qualitatively correct spectral function.

Our aim here is to present numerical methods for solving a general nonlinear programming problem. These methods are based on transformation of a given constrained minimization problem into an unconstrained maximin problem. This transformation is done by using a generalized Lagrange multiplier technique. Such an approach permits us to use Newton's and gradient methods for nonlinear programming. Convergence proofs are provided, and some numerical results are given.

This paper studies the problem of computing an average (or mean) displacement from a set of given spatial displacements using three types of parametric representations: Euler angles and translation vectors, unit quaternions and translation vectors, and dual quaternions. It is shown that the use of Euclidean norm in the space of unit quaternions reduces the problem to that of computing the mean for each quaternion component separately and independently. While the resulting algorithm is simple, a change in the sign of a unit quaternion could lead to an incorrect result. A novel kinematic measure based on dual quaternions is introduced to capture the separation between two spatial displacements. This kinematic measure is used to define the variance of a set of displacements, which is then used to formulate a constrained least squares minimization problem. It is shown that the problem decomposes into that of finding the optimal translation vector and the optimal unit quaternion. The former is simply the centroid of the set of translation vectors and the latter is obtained as the eigenvector corresponding to the least eigenvalue of a 4 × 4 positive definite symmetric matrix. In addition, it is found that the weight factor used in combining rotations and translations in the formulation does not play a role in the final outcome. Examples are provided to show the comparisons of these methods.

ABSTRACTThis paper is concerned with the N coupled nonlinear fractional Schrödinger equations (). The variations and improved concentration-compactness principle are employed to obtain the existence and uniqueness of the ground states. First, we estimate the constrained minimization problem . Next, we prove the subadditivity of which is partial constrained minimization problem defined in the Section 3. Then, we obtain the minimizing sequence cannot be dichotomous. Using the harmonic extension technique, it obtains that the ground states are positive. In addition, the orbital stability of the positive ground states is studied. Our work improves the results of Ohta.

We use the penalty approach to study two constrained minimization problems in finite-dimensional spaces. A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. We use the notion of the Mordukhovich basic subdifferential and establish a simple sufficient condition for the exact penalty property.