Quadratic Minimization Problem Research Articles

A Stable Solution of a Nonuniformly Perturbed Quadratic Minimization Problem by the Extragradient Method with Step Size Separated from Zero

A Stable Solution of a Nonuniformly Perturbed Quadratic Minimization Problem by the Extragradient Method with Step Size Separated from Zero

Nonlinear Gradient Field Estimation in Diffusion MRI Tensor Simulation.

Gradient nonlinearities not only induce spatial distortion in magnetic resonance imaging (MRI), but also introduce discrepancies between intended and acquired diffusion sensitization in diffusion weighted (DW) MRI. Advances in scanner performance have increased the importance of correcting gradient nonlinearities. The most common approaches for gradient nonlinear field estimations rely on phantom calibration field maps which are not always feasible, especially on retrospective data. Here, we derive a quadratic minimization problem for the complete gradient nonlinear field (L(r)). This approach starts with corrupt diffusion signal and estimates the L(r) in two scenarios: (1) the true diffusion tensor known and (2) the true diffusion tensor unknown (i.e., diffusion tensor is estimated). We show the validity of this mathematical approach, both theoretically and through tensor simulation. The estimated field is assessed through diffusion tensor metrics: mean diffusivity (MD), fractional anisotropy (FA), and principal eigenvector (V1). In simulation with 300 diffusion tensors, the study shows the mathematical model is not ill-posed and remains stable. We find when the true diffusion tensor is known (1) the change in determinant of the estimated L(r) field and the true field is near zero and (2) the median difference in estimated L(r) corrected diffusion metrics to true values is near zero. We find the results of L(r) estimation are dependent on the level of L(r) corruption. This work provides an approach to estimate gradient field without the need for additional calibration scans. To the best of our knowledge, the mathematical derivation presented here is novel.

A Block Lanczos Method for Large-Scale Quadratic Minimization Problems with Orthogonality Constraints

A Block Lanczos Method for Large-Scale Quadratic Minimization Problems with Orthogonality Constraints

Second-order cone programming for frictional contact mechanics using interior point algorithm

We report experiments of an implementation of a primal–dual interior point algorithm for solving mechanical models of one-sided contact problems with Coulomb friction. The objective is to recover an optimal solution with high accuracy and as quickly as possible. These developments are part of the design of Siconos, an open-source software for modelling and simulating non-smooth dynamical systems. Currently, Siconos uses mainly first-order methods for the numerical solution of these systems. These methods are very robust, but suffer from a linear rate of convergence and are therefore too slow to find accurate solutions in a reasonable time. Our main objective is to apply second-order methods to speed up convergence. In this paper, we focus on solving a relaxed formulation of the initial mechanical model, corresponding to the optimality conditions of a convex quadratic minimization problem with second-order cone constraints. We will present in detail a primal–dual interior point algorithm for solving this type of problem. The main difficulty in implementing this algorithm arises from the fact that at each iteration of the algorithm, a change of variable, called a scaling, has to be performed in order to guarantee the non-singularity of the linear system to be solved, as well as to recover a symmetric system. Although this scaling strategy is very nice from a theoretical point of view, it leads to an enormous deterioration of the conditioning of the linear system when approaching the optimal solution and therefore to all the numerical difficulties that result from it. We will describe in detail the numerical algebra that we have developed in our implementation, in order to overcome these problems of numerical instability. We will also present the solution of the models resulting from the problems of rolling friction, for which the cone of constraints is no longer self-dual like the Lorentz cone.

New gradient methods with adaptive stepsizes by approximate models

As we know, the stepsize is extremely crucial to gradient method. A new type of stepsize is introduced for the gradient method in the paper, which is generated by minimizing the norm of the approximate model of the gradient along the line of negative gradient direction. Based on the retard technique, we present a new gradient method by minimizing adaptively the approximate model of the objective function and the norm of the approximate model of the gradient along the line of negative gradient direction for strictly convex quadratic minimization. The convergence of the proposed method is established. The numerical experiments on four groups of strictly convex quadratic minimization problems illustrate that the proposed method is very promising. We also extend the new gradient method for convex quadratic minimization to general unconstrained optimization by incorporating a nonmonotone line search. The convergence of the resulting method is established. The numerical experiment on the 147 test functions from the CUTEst library indicates that the resulting method is superior to some efficient gradient methods including the BBQ method (SIAM J Optim. 2021;31(4):3068–3096) and is competitive to two famous conjugate gradient software packages CGOPT (1.0) and CG _ DESCENT (5.0).

Comparing solution paths of sparse quadratic minimization with a Stieltjes matrix

This paper studies several solution paths of sparse quadratic minimization problems as a function of the weighing parameter of the bi-objective of estimation loss versus solution sparsity. Three such paths are considered: the “ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document}-path” where the discontinuous ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document}-function provides the exact sparsity count; the “ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path” where the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-function provides a convex surrogate of sparsity count; and the “capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path” where the nonconvex nondifferentiable capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-function aims to enhance the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-approximation. Serving different purposes, each of these three formulations is different from each other, both analytically and computationally. Our results deepen the understanding of (old and new) properties of the associated paths, highlight the pros, cons, and tradeoffs of these sparse optimization models, and provide numerical evidence to support the practical superiority of the capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path. Our study of the capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path is interesting in its own right as the path pertains to computable directionally stationary (= strongly locally minimizing in this context, as opposed to globally optimal) solutions of a parametric nonconvex nondifferentiable optimization problem. Motivated by classical parametric quadratic programming theory and reinforced by modern statistical learning studies, both casting an exponential perspective in fully describing such solution paths, we also aim to address the question of whether some of them can be fully traced in strongly polynomial time in the problem dimensions. A major conclusion of this paper is that a path of directional stationary solutions of the capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-regularized problem offers interesting theoretical properties and practical compromise between the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document}-path and the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path. Indeed, while the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document}-path is computationally prohibitive and greatly handicapped by the repeated solution of mixed-integer nonlinear programs, the quality of ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path, in terms of the two criteria—loss and sparsity—in the estimation objective, is inferior to the capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path; the latter can be obtained efficiently by a combination of a parametric pivoting-like scheme supplemented by an algorithm that takes advantage of the Z-matrix structure of the loss function.

A multigrid technique for Lagrange multiplier-based unfitted Finite Element contact issues.

Internal interfaces in a domain could exist as a material defect or they can appear due to propagations of cracks. Discretization of such geometries and solution of the contact problem on the internal interfaces can be computationally challenging. We employ an unfitted Finite Element (FE) framework for the discretization of the domains and develop a tailored, globally convergent, and efficient multigrid method for solving contact problems on the internal interfaces. In the unfitted FE methods, structured background meshes are used and only the underlying finite element spaces are modified to incorporate the discontinuities. The non-penetration conditions on the embedded interfaces of the domains are discretized using the method of Lagrange multipliers. We reformulate the arising variational inequality problem as a quadratic minimization problem with linear inequality constraints. Our multigrid method can solve such problems by employing a tailored multilevel hierarchy of the FE spaces and a novel approach for tackling the discretized non-penetration conditions. We employ pseudo-L2 projection- based transfer operators to construct a hierarchy of nested FE spaces from the hierarchy of non-nested meshes. The essential component of our multigrid method is a technique that decouples the linear constraints using an orthogonal transformation. The decoupled constraints are handled by a modified variant of the projected Gauss–Seidel method, which we employ as a smoother in the multigrid method. These components of the multigrid method allow us to enforce linear constraints locally and ensure the global convergence. We will demonstrate the robustness, efficiency, and level independent convergence property of the proposed method for Signorini’s problem and two-body contact problems

Lensfree on-chip microscopy based on single-plane phase retrieval.

We propose a novel single-plane phase retrieval method to realize high-quality sample reconstruction for lensfree on-chip microscopy. In our method, complex wavefield reconstruction is modeled as a quadratic minimization problem, where total variation and joint denoising regularization are designed to keep a balance of artifact removal and resolution enhancement. In experiment, we built a 3D-printed field-portable platform to validate the imaging performance of our method, where resolution chart, dynamic target, transparent cell, polystyrene beads, and stained tissue sections are employed for the imaging test. Compared to state-of-the-art methods, our method eliminates image degradation and obtains a higher imaging resolution. Different from multi-wavelength or multi-height phase retrieval methods, our method only utilizes a single-frame intensity data record to accomplish high-fidelity reconstruction of different samples, which contributes a simple, robust, and data-efficient solution to design a resource-limited lensfree on-chip microscope. We believe that it will become a useful tool for telemedicine and point-of-care application.

A generalized multigrid method for solving contact problems in Lagrange multiplier based unfitted Finite Element method

Internal interfaces in a domain could exist as a material defect or they can appear due to propagations of cracks. Discretization of such geometries and solution of the contact problem on the internal interfaces can be computationally challenging. We employ an unfitted Finite Element (FE) framework for the discretization of the domains and develop a tailored, globally convergent, and efficient multigrid method for solving contact problems on the internal interfaces. In the unfitted FE methods, structured background meshes are used and only the underlying finite element spaces are modified to incorporate the discontinuities. The non-penetration conditions on the embedded interfaces of the domains are discretized using the method of Lagrange multipliers. We reformulate the arising variational inequality problem as a quadratic minimization problem with linear inequality constraints. Our multigrid method can solve such problems by employing a tailored multilevel hierarchy of the FE spaces and a novel approach for tackling the discretized non-penetration conditions. We employ pseudo-L2 projection-based transfer operators to construct a hierarchy of nested FE spaces from the hierarchy of non-nested meshes. The essential component of our multigrid method is a technique that decouples the linear constraints using an orthogonal transformation. The decoupled constraints are handled by a modified variant of the projected Gauss–Seidel method, which we employ as a smoother in the multigrid method. These components of the multigrid method allow us to enforce linear constraints locally and ensure the global convergence. We will demonstrate the robustness, efficiency, and level independent convergence property of the proposed method for Signorini’s problem and two-body contact problems.

Support Solving Method for Box-constrained Indefinite Quadratic Minimization Problems

Support Solving Method for Box-constrained Indefinite Quadratic Minimization Problems

Stable Solution of a Quadratic Minimization Problem with a Nonuniformly Perturbed Operator by Applying a Regularized Gradient Method

Stable Solution of a Quadratic Minimization Problem with a Nonuniformly Perturbed Operator by Applying a Regularized Gradient Method

Sketching for Elimination of Communication Links in LQG Teams

We consider here a scenario where a team of agents want to switch from a fully connected communication network to a network with limited bandwidth because of a cyber attack. The goal is to identify which communication link should still be active in the new network for near-optimal overall performance. This is formulated as a cardinality constrained quadratic minimization problem, which is NP-hard in general. To obtain an approximately optimal solution efficiently, we used random projection (RP) for dimensionality reduction. We show that the value of the sketched team is bounded above by a constant factor times the optimal value of the team with high probability.

Adaptive Global Algorithm for Solving Box-Constrained Non-convex Quadratic Minimization Problems

Adaptive Global Algorithm for Solving Box-Constrained Non-convex Quadratic Minimization Problems

On a Quadratic Minimization Problem with Nonuniform Perturbations in the Criteria and Constraints

On a Quadratic Minimization Problem with Nonuniform Perturbations in the Criteria and Constraints

Joint energy and capacity equilibrium model for centralized and behind-the-meter distributed generation

This paper presents a conjectured-price-response equilibrium approach for modeling both centralized generation (CG) and behind-the-meter distributed generation (BMDG). A Nash game is set up with two constraints linking the CG and BMDG decisions to satisfy both the electricity demand in an energy market and the firm capacity in a capacity market. CG agents maximize their market profits while BMDG customers minimize their net supply costs, making decisions on their annual capacity investments and hourly productions decisions. Customers’ costs account for 1) the energy bought from the grid minus the BMDG energy surpluses sold; 2) the payment of the grid access tariff (power and energy-based terms) and 3) the BMDG capacity investments’ costs. The equilibrium conditions enable to represent different degrees of oligopoly using conjectural variations in both the energy and capacity markets. This work proves that such an equilibrium problem can be solved through an equivalent, yet simpler-to-solve, quadratic minimization problem. Some case examples compare the results of the proposed joint energy and capacity equilibrium with those from an energy-only equilibrium. Among other conclusions, these cases show that the proposed equilibrium sends adequate economic signals to the consumers to taper off the total system peak demand, whenever the weight of the power-based term of the access tariff is not extremely high.

Minimization with differential inequality constraints applied to complete Lyapunov functions

Motivated by the desire to compute complete Lyapunov functions for nonlinear dynamical systems, we develop a general theory of discretizing a certain type of continuous minimization problems with differential inequality constraints. The resulting discretized problems are quadratic optimization problems, for which there exist efficient solution algorithms, and we show that their unique solutions converge strongly in appropriate Sobolev spaces to the unique solution of the original continuous problem. We develop the theory and present examples of our approach, where we compute complete Lyapunov functions for nonlinear dynamical systems. A complete Lyapunov function characterizes the behaviour of a general dynamical system. In particular, the state space is divided into the chain-recurrent set, where the complete Lyapunov function is constant along solutions, and the part characterizing the gradient-like flow, where the complete Lyapunov function is strictly decreasing along solutions. We propose a new method to compute a complete Lyapunov function as the solution of a quadratic minimization problem, for which no information about the chain-recurrent set is required. The solutions to the discretized problems, which can be solved using quadratic programming, converge to the complete Lyapunov function.

A Primal-Dual Projection Algorithm for Efficient Constraint Preconditioning

We consider a linear iterative solver for large scale linearly constrained quadratic minimization problems that arise, for example, in optimization with PDEs. By a primal-dual projection (PDP) iteration, which can be interpreted and analysed as a gradient method on a quotient space, the given problem can be solved by computing sulutions for a sequence of constrained surrogate problems, projections onto the feasible subspaces, and Lagrange multiplier updates. As a major application we consider a class of optimization problems with PDEs, where PDP can be applied together with a projected cg method using a block triangular constraint preconditioner. Numerical experiments show reliable and competitive performance for an optimal control problem in elasticity.

Quadratic problems with two quadratic constraints: convex quadratic relaxation and strong lagrangian duality

In this paper, we study a nonconvex quadratic minimization problem with two quadratic constraints, one of which being convex. We introduce two convex quadratic relaxations (CQRs) and discuss cases, where the problem is equivalent to exactly one of the CQRs. Particularly, we show that the global optimal solution can be recovered from an optimal solution of the CQRs. Through this equivalence, we introduce new conditions under which the problem enjoys strong Lagrangian duality, generalizing the recent condition in the literature. Finally, under the new conditions, we present necessary and sufficient conditions for global optimality of the problem.

A hybrid gradient method for strictly convex quadratic programming

AbstractIn this article, we present a reliable hybrid algorithm for solving convex quadratic minimization problems. At thekth iteration, two points are computed: first, an auxiliary point is generated by performing a gradient step using an optimal steplength, and second, the next iteratexk + 1is obtained by means of weighted sum of with the penultimate iteratexk − 1. The coefficient of the linear combination is computed by minimizing the residual norm along the line determined by the previous points. In particular, we adopt an optimal, nondelayed steplength in the first step and then use a smoothing technique to impose a delay on the scheme. Under a modest assumption, we show that our algorithm is Q‐linearly convergent to the unique solution of the problem. Finally, we report numerical experiments on strictly convex quadratic problems, showing that the proposed method is competitive in terms of CPU time and iterations with the conjugate gradient method.

Robustness Analysis of a Power-Type Varying-Parameter Recurrent Neural Network for Solving Time-Varying QM and QP Problems and Applications

Varying-parameter recurrent neural network, being a special kind of neural-dynamic methodology, has revealed powerful abilities to handle various time-varying problems, such as quadratic minimization (QM) and quadratic programming (QP) problems. In this paper, a novel power-type varying-parameter recurrent neural network (PT-VP-RNN) is proposed to solve the perturbed time-varying QM and QP problems. First, based on the generalization of time-varying QM and QP problems, the design process of the PT-VP-RNN is presented in detail. Second, the robustness performance of the proposed PT-VP-RNN is theoretically analyzed and proved. What is more, two numerical examples are simulated to illustrate the robustness convergence performance of PT-VP-RNN even in a large disturbance condition. Finally, two practical application examples (i.e., a robot tracking example and a venture investment example) further verify the effectiveness, accuracy, and widespread applicability of the proposed PT-VP-RNN.