Constrained Minimization Problem Research Articles

A new algorithm for computing the nearest polynomial to multiple given polynomials via weighted ℓ2,q-norm minimization and its complex extension

A new algorithm is proposed in this paper for computing the nearest polynomial to multiple given polynomials with a given zero in the real case, where the distance between polynomials is defined by the weighted ℓ2,q norm (0<q≤2). First, the problem is formulated as a univariate constrained non-convex minimization problem, where prior information of the coefficients of the polynomial can be embedded by selecting proper weights. Then, an iteratively reweighted algorithm is designed to solve the obtained problem, and also the convergence and rate of convergence are uniformly demonstrated for all q in (0,2]. Since all the existing methods for computing the nearest polynomial to multiple given polynomials with a given zero are limited to the real case, we ingeniously extend the results to the complex case. Finally, two representative examples that separately compute the nearest real and complex polynomials are presented to show the effectiveness of the proposed algorithm.

Accurate computing paradigm for decoupled harmonic load flow analysis with CONOPT solvers for non-sinusoidal radial distribution systems

In this article, a novel method for performing Decoupled Harmonic Load Flow (DHLF) analysis in radial distribution systems is presented. The proposed method involves solving load flow equations for both fundamental and higher frequencies as constrained minimization problems utilizing a nonlinear programming (NLP) CONOPT solver integrated into the General Algebraic Modeling Systems (GAMS) environment. The effectiveness of the proposed method is assessed across three IEEE test systems: 18-bus, 33-bus, and 69-bus. Five cases of DHLF analysis are taken into consideration. DHLF was performed on standard test systems with incorporated non-linear loads in the first case. Cases two through five examine the DHLF in the presence of distributed generators (DGs), shunt capacitors (SCs), simultaneous DGs and SCs, and electric vehicles (EVs). Comparing the obtained results to those of DHLF analysis tools ETAP and DigSilent, as well as to those found in the literature, proved that the proposed CONOPT solver provides precise solutions.

Some Theoretical Foundations of Bare-Simulation Optimization of Some Directed Distances between Fuzzy Sets Respectively Basic Belief Assignments.

It is well known that in information theory-as well as in the adjacent fields of statistics, machine learning and artificial intelligence-it is essential to quantify the dissimilarity between objects of uncertain/imprecise/inexact/vague information; correspondingly, constrained optimization is of great importance, too. In view of this, we define the dissimilarity-measure-natured generalized φ-divergences between fuzzy sets, ν-rung orthopair fuzzy sets, extended representation type ν-rung orthopair fuzzy sets as well as between those fuzzy set types and vectors. For those, we present how to tackle corresponding constrained minimization problems by appropriately applying our recently developed dimension-free bare (pure) simulation method. An analogous program is carried out by defining and optimizing generalized φ-divergences between (rescaled) basic belief assignments as well as between (rescaled) basic belief assignments and vectors.

Existence and Limit Behavior of Constraint Minimizers for a Varying Non-Local Kirchhoff-Type Energy Functional

In this paper, we study the constrained minimization problem for an energy functional which is related to a Kirchhoff-type equation. For s=1, there many articles have analyzed the limit behavior of minimizers when η&gt;0 as b→0+ or b&gt;0 as η→0+. When the equation involves a varying non-local term ∫R3|∇u|2dxs, we give a detailed limit behavior analysis of constrained minimizers for any positive sequence {ηk} with ηk→0+. The present paper obtains an interesting result on this topic and enriches the conclusions of previous works.

Variance Reduced Random Relaxed Projection Method for Constrained Finite-Sum Minimization Problems

Variance Reduced Random Relaxed Projection Method for Constrained Finite-Sum Minimization Problems

Generalized nesterov’s algorithm for constrained minimization problems on closed convex sets

In this paper, we propose a generalized Nesterov algorithm for the constrained optimization problems on a closed convex set. We prove the convergence as well as the convergence rate of the proposed algorithm. First, we present a new algorithm based on the generalization of Nesterov’s algorithm. Then, we prove the convergence as well as the convergence rate of the new algorithm. With a specific choice of parameters, the new algorithm becomes Nesterov’s algorithm. Therefore, the convergence as well as the convergence rate of Nesterov’s algorithm are also followed. We illustrate the effectiveness of the new algorithm as well as compare it with Nesterov’s algorithm and the gradient descent algorithm through a specific example.

Viscoacoustic least squares reverse-time migration using L1-2 norm sparsity constraint

Abstract Least-squares reverse-time migration (LSRTM) has become an advanced technique for complex structures imaging of the subsurface, as it can provide a higher resolution and more balanced amplitude migrated image than conventional reverse-time migration (RTM). However, the intrinsic attenuation of subsurface introduces amplitude attenuation and phase dispersion of seismic wavefield, which leads to the inverted image kinematically and dynamically inexactitude. Moreover, the imperfect geometry, limited bandwidth of seismic data, and inappropriate modeling kernel etc., would inevitably introduce two side-effects in migrated image, resulting in degradation of LSRTM imaging potential. To alleviate above issues, we present a data-domain sparsity constraint viscoacoustic least-squares reverse-time migration algorithm in this paper. In particular, we utilize the decoupled constant Q fractional Laplacians (DFLs) viscoacoustic wave equation as the modeling kernel to describe the attenuation effects of the subsurface, while a model constraint constructed in the misfit function via L1-2 norm is carried out to clear the migrated artefacts and boost the imaging resolution. Thanks to the excellent performance in sparsity, the drawbacks of unconstraint LSRTM can be effectively mitigated by the L1-2 norm-based regularization. In this paper, we adopt the alternating direction of multipliers method (ADMM) to iteratively address the constrained L1-2 minimization problem by implementing a proximal operator, and three synthetic examples are hired to evaluate the effectiveness and practicability of the proposed strategy. Migration results prove that the proposed scheme can effectively compensate the attenuation effects, improve the resolution, and suppress the migration artifacts of inverted images even in the complex imaging situations.

A Regularized High-Dimensional Positive Definite Covariance Estimator with High-Frequency Data

This paper proposes a novel large-dimensional positive definite covariance estimator for high-frequency data under a general factor model framework. We demonstrate an appealing connection between the proposed estimator and a weighted group least absolute shrinkage and selection operator (LASSO) penalized least-squares estimator. The proposed estimator improves on traditional principal component analysis by allowing for weak factors, whose signal strengths are weak relative to idiosyncratic components. Despite the presence of microstructure noises and asynchronous trading, the proposed estimator achieves guarded positive definiteness without sacrificing the convergence rate. To make our method fully operational, we provide an extended simultaneous alternating direction method of multipliers algorithm to solve the resultant constrained convex minimization problem efficiently. Empirically, we study the monthly high-frequency covariance structure of the stock constituents of the S&amp;P 500 index from 2008 to 2016, using all traded stocks from the NYSE, AMEX, and NASDAQ stock markets to construct the high-frequency Fama-French four and extended eleven economic factors. We further examine the out-of-sample performance of the proposed method through vast portfolio allocations, which deliver significantly reduced out-of-sample portfolio risk and enhanced Sharpe ratios. The success of our method supports the usefulness of machine learning techniques in finance. This paper was accepted by Agostino Capponi, finance. Funding: This work was supported by the Research Grants Council, University Grants Committee [Grants 11500119, 11505522, 11505721, and 21504818] and the National Natural Science Foundation of China (NSFC) Basic Scientific Center Project [Grant 71988101], entitled as “Econometric Modelling and Economic Policy Studies”, as well as NSFC [Grants 71803166 and 72173104]. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2022.04138 .

An augmented Lagrangian approach for cardinality constrained minimization applied to variable selection problems

To solve convex constrained minimization problems, that also include a cardinality constraint, we propose an augmented Lagrangian scheme combined with alternating projection ideas. Optimization problems that involve a cardinality constraint are NP-hard mathematical programs and typically very hard to solve approximately. Our approach takes advantage of a recently developed and analyzed continuous formulation that relaxes the cardinality constraint. Based on that formulation, we solve a sequence of smooth convex constrained minimization problems, for which we use projected gradient-type methods. In our setting, the convex constraint region can be written as the intersection of a finite collection of convex sets that are easy and inexpensive to project. We apply our approach to a variety of over and under determined constrained linear least-squares problems, with both synthetic and real data that arise in variable selection, and demonstrate its effectiveness.

Reconstruction regions for unobserved middle order statistics in the exponential distribution

There are some situations in reliability studies in which a middle part of the sample is missing. For example, in life-testing experiments, the experimenter may not enable to observe the middle lifetimes. This may be due to negligence, removal of test units (censoring) and or equipment failure and technical problems in the middle of the experiment. In this paper, the problem of reconstructing unobserved middle lifetimes from the exponential distribution is discussed and reconstruction regions (or sets) are obtained. First, it is shown how to find balanced reconstruction regions for the unobserved middle lifetimes. Then, a constrained minimization problem is used to find the reconstruction region with the smallest area. For illustrative and comparative purposes, a simulation study and two numerical examples are presented.

A hybrid direct search and projected simplex gradient method for convex constrained minimization

We propose a new Derivative-free Optimization (DFO) approach for solving convex constrained minimization problems. The feasible set is assumed to be the non-empty intersection of a finite collection of closed convex sets, such that the projection onto each of these individual convex sets is simple and inexpensive to compute. Our iterative approach alternates between steps that use Directional Direct Search (DDS), considering adequate poll directions, and a Spectral Projected Gradient (SPG) method, replacing the real gradient by a simplex gradient, under a DFO approach. In the SPG steps, if the convex feasible set is simple, then a direct projection is computed. If the feasible set is the intersection of finitely many convex simple sets, then Dykstra's alternating projection method is applied. Convergence properties are established under standard assumptions usually associated to DFO methods. Some preliminary numerical experiments are reported to illustrate the performance of the proposed algorithm, in particular by comparing it with a classical DDS method. Our results indicate that the hybrid algorithm is a robust and effective approach for derivative-free convex constrained optimization.

Traffic Count Estimation at Basis Links Without Path Flow and Historic Data

Traffic counts (or link counts) are defined as cumulative traffic in the lanes between two consecutive intersections on a road network. Established methods of link count estimation assume the availability of count data at a set of basis links: a minimum subset of all the links of a network that still allow complete network traffic count estimation. If traffic count data are missing even at some basis links, current research must introduce additional assumptions, on path flow or historical data, to compensate. In this research, we present an approach to estimate the missing basis link count without the need for historical data or path-flow information, thereby overcoming the limitations of state-of-the-art estimation approaches. We develop a stochastic method using a canonical correlation analysis-based constrained minimization problem for estimation purposes. The proposed method has been validated with real-world link count data collected in Melbourne, Australia, between <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2016$</tex-math> </inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2019$</tex-math> </inline-formula> . The validation results indicate that we can achieve an accuracy of up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$90\%$</tex-math> </inline-formula> in the real world, despite the unknown traffic patterns of the estimation period. Depending on the time of day, the modelling strategy selected, and the consistency of input data available at the road intersection, the estimation accuracy varies. The proposed methodology is useful when there is a general shortage of data since there is inadequate infrastructure for data collection in many major cities around the world.

Proximal linearized alternating direction method of multipliers algorithm for nonconvex image restoration with impulse noise

AbstractImage restoration with impulse noise is an important task in image processing. Taking into account the statistical distribution of impulse noise, the ℓ1‐norm data fidelity and total variation () model has been widely used in this area. However, the model usually performs worse when the noise level is high. To overcome this drawback, several nonconvex models have been proposed. In this paper, an efficient iterative algorithm is proposed to solve nonconvex models arising in impulse noise. Compared to existing algorithms, the proposed algorithm is a completely explicit algorithm in which every subproblem has a closed‐form solution. The key idea is to transform the original nonconvex models into an equivalent constrained minimization problem with two separable objective functions, where one is differentiable but nonconvex. As a consequence, the proximal linearized alternating direction method of multipliers is employed to solve it. Extensive numerical experiments are presented to demonstrate the efficiency and effectiveness of the proposed algorithm.

Homogenization and Phase Separation with Space Dependent Wells: The Subcritical Case

A variational model for the interaction between homogenization and phase separation is considered. The focus is on the regime where the latter happens at a smaller scale than the former, and when the wells of the double well potential are allowed to move and to have discontinuities. The zeroth and first order Gamma -limits are identified. The topology considered for the latter is that of two-scale, since it encodes more information on the asymptotic local microstructure. In particular, when the wells are non-constant, the first order Gamma -limit describes the contribution of microscopic phase separation, even in situations where there is no macroscopic phase separation. As a corollary, the minimum of the mass constrained minimization problem is characterized, and it is shown to depend on whether or not the wells are discontinuous. In the process of proving these results, the theory of inhomogeneous Modica Mortola functionals is strengthened.

Properties of minimizers for the fractional Kirchhoff energy functional

In this paper, we are concerned with a fractional Kirchhoff equation with a general coercive potential. First, we consider some existence and nonexistence of L2-constraint minimizers for related constrained minimization problems. Most importantly, by constructing appropriate trial functions for some delicate energy estimates and studying decay properties of solution sequences, we then establish the concentration behaviors of L2-constraint minimizers for related constrained minimization problems.

Variance reduced moving balls approximation method for smooth constrained minimization problems

Variance reduced moving balls approximation method for smooth constrained minimization problems

Existence of coupled optical vortex solitons propagating in a quadratic nonlinear medium

We consider the coupled propagation of an optical field and its second harmonic in a quadratic nonlinear medium governed by a coupled system of Schrodinger equations. We prove the existence of ring‐profiled optical vortex solitons appearing as solutions to a constrained minimization problem and as solutions to a min–max problem. In the case of the constrained minimization problem, solutions are shown to be positive with undetermined wave propagation constants, but in the min–max approach, the wave propagation constants can be prescribed. The quadratic nonlinearity introduces some interesting properties not commonly observed in other coupled systems in the context of nonlinear optics, such as the system not accepting any semi‐trivial solutions, meaning that optical solitons cannot be observed when, say, one of the beams is off. Additionally, the second harmonic always remains positive.

Pattern-Multiplicative Average of Nonnegative Matrices Revisited: Eigenvalue Approximation Is the Best of Versatile Optimization Tools

Given several nonnegative matrices with a single pattern of allocation among their zero/nonzero elements, the average matrix should have the same pattern, too. This is the first tenet of the pattern-multiplicative average (PMA) concept, while the second one suggests the multiplicative (or geometric) nature of averaging. The original concept of PMA was motivated by the practice of matrix population models as a tool to assess the population viability from long-term monitoring data. The task has reduced to searching for an approximate solution to an overdetermined system of polynomial equations for unknown elements of the average matrix (G), and hence to a nonlinear constrained minimization problem for the matrix norm. Former practical solutions faced certain technical problems, which required sophisticated algorithms but returned acceptable estimates. Now, we formulate (for the first time in ecological modeling and nonnegative matrix theory) the PMA problem as an eigenvalue approximation one and reduce it to a standard problem of linear programing (LP). The basic equation of averaging also determines the exact value of λ1(G), the dominant eigenvalue of matrix G, and the corresponding eigenvector. These are bound by the well-known linear equations, which enable an LP formulation of the former nonlinear problem. The LP approach is realized for 13 fixed-pattern matrices gained in a case study of Androsace albana, an alpine short-lived perennial, monitored on permanent plots over 14 years. A standard software routine reveals the unique exact solution, rather than an approximate one, to the PMA problem, which turns the LP approach into ‘’the best of versatile optimization tools”. The exact solution turns out to be peculiar in reaching zero bounds for certain nonnegative entries of G, which deserves modified problem formulation separating the lower bounds from zero.

On Coupled Two-Level Variational Problem in Sobolev-Orlicz Space

We study a coupled two-level variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its consistence and solvability issues. At the first level, we deal with the so-called temporal interpolation problem that can be cast as a state constrained optimal control problem for anisotropic convection-diffusion equation with two types of control functions — distributed $L^2$-control and $BV$-bounded control in coefficients. At the second level, we have a constrained minimization problem with the nonstandard growth energy functional that lives in a variable Sobolev-Orlicz space. The characteristic feature of the proposed model is the fact that the variable exponent, which is associated with non-standard growth in the objective functional, is unknown a priori and it depends on the solution of the first-level optimal control problem.

Properties of the minimizers for a constrained minimization problem arising in fractional NLS system

Properties of the minimizers for a constrained minimization problem arising in fractional NLS system