Minimization Problem Research Articles

Elastic Localization With Particular Reference to Tape-Springs

Abstract Many elastic systems localize under applied displacement, precipitating into regions of lower and higher strain; further displacement is accommodated by growth of the high strain region at a constant load. Such systems can be studied as propagating instabilities, focusing on the work required to propagate the high strain region, or as two-phase energy minimization problems. It is shown that the Maxwell “equal-areas” construction, and the related common tangent construction, provide the solution to either approach. A new, graphical, proof of the Maxwell equal-areas construction using total strain energy diagrams is presented. Tape-springs are investigated as a case study, with localization presenting as the formation of elastic folds—developable regions with high curvature. One notable property of tape-spring folds is that the fold radius is approximately equal to the initial transverse radius. This result was first proven by Rimrott, and later improved by Calladine and Seffen. A further improvement is obtained here by application of the common tangent construction, and all solutions are shown to be approximations to the Maxwell equal-areas construction in the limit of zero thickness.

A Neural Network Approach for Solving Weighted Nonlinear Complementarity Problems

In this paper, we consider the weighted nonlinear complementarity problem which is the extension of the general complementarity problems and contains a wide class of optimization problems. Based on the weighted complementarity function, we reformulate the weighted nonlinear complementarity problem as an unconstrained minimization problem and present the steepest descent-based neural network to solve it. Under mild conditions, we derive some results related to the asymptotic stability of the neural network. Numerical experiments indicate that the proposed algorithm is quite effective.

Signal and image reconstruction with a double parameter Hager–Zhang‐type conjugate gradient method for system of nonlinear equations

AbstractThe one parameter conjugate gradient method by Hager and Zhang (Pac J Optim, 2(1):35–58, 2006) represents a family of descent iterative methods for solving large‐scale minimization problems. The nonnegative parameter of the scheme determines the weight of conjugacy and descent, and by extension, the numerical performance of the method. The scheme, however, does not converge globally for general nonlinear functions, and when the parameter approaches 0, the scheme reduces to the conjugate gradient method by Hestenes and Stiefel (J Res Nat Bur Stand, 49:409–436, 1952), which in practical sense does not perform well due to the jamming phenomenon. By carrying out eigenvalue analysis of an adaptive two parameter Hager–Zhang type method, a new scheme is presented for system of monotone nonlinear equations with its application in compressed sensing. The proposed scheme was inspired by nice attributes of the Hager–Zhang method and the various schemes designed with double parameters. The scheme is also applicable to nonsmooth nonlinear problems. Using fundamental assumptions, analysis of the global convergence of the scheme is conducted and preliminary report of numerical experiments carried out with the scheme and some recent methods indicate that the scheme is promising.

Robust adaptive beamforming for cylindrical uniform conformal arrays based on low-rank covariance matrix reconstruction

Recently, conformal arrays have attracted considerable interest because such arrays can provide reduced radar cross-section and increased angle coverage. In this article, we devise a robust adaptive beamforming (RAB) approach using cylindrical uniform conformal array (CUCA). Firstly, we derive the minimum variance distortionless response (MVDR) beamformer for the CUCA by utilizing the noise subspace of interference covariance matrix (ICM) and steering vector (SV) of the signal-of-interest (SOI). Subsequently, the ICM is reconstructed by estimating the noise-free covariance matrix of the CUCA outputs and the interference projection matrix. Specifically, the noise-free covariance matrix can be regarded as multiple low-rank covariance matrices, and each low-rank matrix is reconstructed by formulating a nuclear norm minimization (NNM) problem. With the reconstructed covariance matrix, the 2-D DOAs of sources are determined by employing 2-D MUSIC spectrum to form the interference projection matrix. In addition, the SOI SV is estimated by solving a quadratically constrained quadratic programming (QCQP) problem. Numerical results demonstrate that the proposed approach is obviously superior to the existing RAB techniques.

Continuation problem of the electromagnetic field in the frequency domain for horizontally layred media

The paper presents analytical expressions for solving the problem of continuation of the electromagnetic field in the frequency domain for a horizontally layered medium. The numerical solution algorithm uses layerwise recalculation of the required quantities, for which analytical expressions are presented in a form that allows calculations and layerwise recalculation without the accumulation of rounding errors. The solution to the continuation problem is obtained as a solution to the cost functional minimization problem. The strong convexity of the functional is proven, which implies the existence of a unique solution to the problem posed.

A note on the 1-D minimization problem related to solenoidal improvement of the uncertainty principle inequality

This paper gives a second way to solve the one-dimensional minimization problem of the form : minf≢0∫0∞f′′2xμ+1dx∫0∞x2f′2-εf2xμ-1dx∫0∞f′2xμdx2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\min _{f\ ot \\equiv 0}\\frac{\\displaystyle \\int \\limits _0^\\infty \\left( f''\\right) ^2x^{\\mu +1}dx\\int \\limits _0^\\infty \\left( {x}^2\\left( f'\\right) ^2 -\\varepsilon f^2\\right) {{x}}^{\\mu -1}d{x}}{\\displaystyle \\left( \\int \\limits _0^\\infty \\left( f'\\right) ^2 {{x}}^{\\mu }d{x}\\right) ^2} \\end{aligned}$$\\end{document}for scalar-valued functions f on the half line, where f′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f'$$\\end{document} and f′′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f''$$\\end{document} are its derivatives and ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document} and μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document} are positive parameters with ε<μ24.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon < \\frac{\\mu ^2}{4}.$$\\end{document} This problem plays an essential part of the calculation of the best constant in Heisenberg’s uncertainty principle inequality for solenoidal vector fields. The above problem was originally solved by using (generalized) Laguerre polynomial expansion; however, the calculation was complicated and long. In the present paper, we give a simpler method to obtain the same solution, the essential part of which was communicated in Theorem 5.1 of the preprint (Hamamoto, arXiv:2104.02351v4, 2021).

Analysis and optimal control of a hierarchical age-structured population model in a polluted environment

For individual biological populations, examining the impacts of competition within the species and the presence of toxic substances in the environment can effectively facilitate control measures. However research in this area remains limited. This article introduces a toxicant-population model incorporating a hierarchical age-structured and examines the minimum deviation problem within this model. The aim is to explore effective control strategies that strike a balance between improving the environment and population benefits. Through a novel approach, we reformulate the problem as an abstract Cauchy problem on a non-dense domain and then apply integrated semigroup theory to establish the existence, uniqueness, and positivity of solutions to the model. The existence of optimal control is derived using Ekeland's variational principle in functional analysis. The expression for optimal control is given by the Pontryagin's minimum principle. Lastly, we conducted numerical experiments on the model, applied optimality conditions to delineate optimal control, and illustrated the comparison between crucial population parameters under optimal control and those without any control.

Minimization of the lowest positive Neumann-Dirichlet eigenvalue for general indefinite Sturm-Liouville problems

The aim of this paper is to obtain the sharp estimate for the lowest positive eigenvalue λ0ND+ for the general Sturm–Liouville problemy″=q(t)y+λm(t)y, with the Neumann-Dirichlet boundary conditions, where q is a nonnegative potential and another potential m admits to change sign. First, we will study the optimal lower bound for the smallest positive eigenvalue in the measure differential equations to make our results more applicable. Second, based on the relationship between the minimization problem of the smallest positive eigenvalue for the ODE and the one for the MDE, we find the explicit optimal lower bound of the smallest positive eigenvalue for the general Sturm–Liouville equation.

A Girsanov transformed Clark-Ocone-Haussmann type formula for $$L^1$$-pure jump additive processes and its application to portfolio optimization

AbstractWe derive a Clark-Ocone-Haussmann (COH) type formula under a change of measure for $$ L^1 $$ L 1 -canonical additive processes, providing a tool for representing financial derivatives under a risk-neutral probability measure. COH formulas are fundamental in stochastic analysis, providing explicit martingale representations of random variables in terms of their Malliavin derivatives. In mathematical finance, the COH formula under a change of measure is crucial for representing financial derivatives under a risk-neutral probability measure. To prove our main results, we use the Malliavin-Skorohod calculus in $$ L^0 $$ L 0 and $$ L^1 $$ L 1 for additive processes, as developed by Di Nunno and Vives (2017). An application of our results is solving the local risk minimization (LRM) problem in financial markets driven by pure jump additive processes. LRM, a prominent hedging approach in incomplete markets, seeks strategies that minimize the conditional variance of the hedging error. By applying our COH formula, we obtain explicit expressions for locally risk-minimizing hedging strategies in terms of Malliavin derivatives under the market model underlying the additive process. These formulas provide practical tools for managing risks in financial market price fluctuations with $$L^1$$ L 1 -additive processes.

Energy Consumption Minimization with SNR Constraint for Wireless Powered Communication Networks.

The article addresses the energy consumption minimization problem in wireless powered communication networks (WPCNs) and proposes a time allocation scheme, named DaTA, which is based on the Different Target Simultaneous Wireless Information and Power Transfer (DT-SWIPT) scheme such that the wireless station can share the remaining energy after transmission to the Hybrid Access Point (HAP) to those who have not transmitted to the HAP to minimize the energy consumption of the WPCN. In addition to proposing a new frame structure, the article also considers the Signal-to-Noise (SNR) constraint to guarantee that the HAP can successfully receive data from wireless stations. In the article, the problem of minimization of energy consumption is formulated as a nonlinear programming model. We employ the SQP (Sequential Quadratic Programming) algorithm to figure out the optimal solution. Moreover, a heuristic method is proposed as well to obtain a near-optimal solution in a shorter time. The simulation results showed that the proposed scheme outperforms the related work in terms of energy consumption and energy efficiency.

Analysis of a Class of Minimization Problems Lacking Lower Semicontinuity

The minimization of nonlower semicontinuous functions is a difficult topic that has been minimally studied. Among such functions is a Heaviside composite function that is the composition of a Heaviside function with a possibly nonsmooth multivariate function. Unifying a statistical estimation problem with hierarchical selection of variables and a sample average approximation of composite chance constrained stochastic programs, a Heaviside composite optimization problem is one whose objective and constraints are defined by sums of possibly nonlinear multiples of such composite functions. Via a pulled-out formulation, a pseudostationarity concept for a feasible point was introduced in an earlier work as a necessary condition for a local minimizer of a Heaviside composite optimization problem. The present paper extends this previous study in several directions: (a) showing that pseudostationarity is implied by (and thus, weaker than) a sharper subdifferential-based stationarity condition that we term epistationarity; (b) introducing a set-theoretic sufficient condition, which we term a local convexity-like property, under which an epistationary point of a possibly nonlower semicontinuous optimization problem is a local minimizer; (c) providing several classes of Heaviside composite functions satisfying this local convexity-like property; (d) extending the epigraphical formulation of a nonnegative multiple of a Heaviside composite function to a lifted formulation for arbitrarily signed multiples of the Heaviside composite function, based on which we show that an epistationary solution of the given Heaviside composite program with broad classes of B-differentiable component functions can in principle be approximately computed by surrogation methods. Funding: The work of Y. Cui was based on research supported by the National Science Foundation [Grants CCF-2153352, DMS-2309729, and CCF-2416172] and the National Institutes of Health [Grant 1R01CA287413-01]. The work of J.-S. Pang was based on research supported by the Air Force Office of Scientific Research [Grant FA9550-22-1-0045].

Dynamic Framing and Power Allocation for Real-Time Wireless Networks with Variable-Length Coding: A Tandem Queue Approach

Ensuring high reliability and low latency poses challenges for numerous applications that require rigid performance guarantees, such as industrial automation and autonomous vehicles. Our research primarily concentrates on addressing the real-time requirements of ultra-reliable low-latency communication (URLLC). Specifically, we tackle the challenge of hard delay constraints in real-time transmission systems, overcoming this obstacle through a finite blocklength coding scheme. In the physical layer, we encode randomly arriving packets using a variable-length coding scheme and transmit the encoded symbols by truncated channel inversion over parallel channels. In the network layer, we model the encoding and transmission processes as tandem queues. These queues backlog the data bits waiting to be encoded and the encoded symbols to be transmitted, respectively. This way, we represent the system as a two-dimensional Markov chain. By focusing on instances when the symbol queue is empty, we simplify the Markov chain into a one-dimensional Markov chain, with the packet queue being the system state. This approach allows us to analytically express power consumption and formulate a power minimization problem under hard delay constraints. Finally, we propose a heuristic algorithm to solve the problem and provide an extensive evaluation of the trade-offs between the hard delay constraint and power consumption.

Multi-objective sand cat swarm optimization based on adaptive clustering for solving multimodal multi-objective optimization problems

Multimodal multi-objective optimization problems (MMOPs) represent a highly challenging class of complex problems, characterized by the presence of several Pareto solution sets in the decision space which map to the identical Pareto-optimal front. The goal of solving MMOPs is to find multiple distinct Pareto sets to sustain a balance between good convergence and diversification of populations. In this paper, a multi-objective sand cat swarm optimization algorithm (MOSCSO) is developed to address MMOPs. In the MOSCSO algorithm, an adaptive clustering-based specific congestion distance technique is introduced to compute the level of crowdedness. This ensures an even distribution of individuals, avoiding excessive crowding in the local area. Subsequently, enhanced search-and-attack prey updating mechanisms are designed to effectively increase not only the exploration and exploitation capabilities of the algorithm but also to enhance the diversity of the swarm in both the decision space and the objective space. To verify the effectiveness of the proposed algorithm, the MOSCSO is applied to solve the CEC2019 complex multimodal benchmark function. The experimental outcomes illustrate that the proposed approach possesses excellent performance in searching for Pareto solutions compared with other algorithms. Meanwhile, the method is also employed to address the map-based distance minimization problem, which further validates the usefulness of the MOSCSO.

Second Order Dynamics Featuring Tikhonov Regularization and Time Scaling

In a Hilbert setting we aim to study a second order in time differential equation, combining viscous and Hessian-driven damping, containing a time scaling parameter function and a Tikhonov regularization term. The dynamical system is related to the problem of minimization of a nonsmooth convex function. In the formulation of the problem as well as in our analysis we use the Moreau envelope of the objective function and its gradient and heavily rely on their properties. We show that there is a setting where the newly introduced system preserves and even improves the well-known fast convergence properties of the function and Moreau envelope along the trajectories and also of the gradient of Moreau envelope due to the presence of time scaling. Moreover, in a different setting we prove strong convergence of the trajectories to the element of minimal norm from the set of all minimizers of the objective. The manuscript concludes with various numerical results.

Majorisation‐minimisation algorithm for optimal state discrimination in quantum communications

AbstractDesigning optimal measurement operators for quantum state discrimination (QSD) is an important problem in quantum communications and cryptography applications. Prior works have demonstrated that optimal quantum measurement operators can be obtained by solving a convex semidefinite program (SDP). However, solving the SDP can represent a high computational burden for many real‐time quantum communication systems. To address this issue, a majorisation‐minimisation (MM)‐based algorithm, called Quantum Majorisation‐Minimisation (QMM) is proposed for solving the QSD problem. In QMM, the authors reparametrise the original objective, then tightly upper‐bound it at any given iterate, and obtain the next iterate as a closed‐form solution to the upper‐bound minimisation problem. Our numerical simulations demonstrate that the proposed QMM algorithm significantly outperforms the state‐of‐the‐art SDP algorithm in terms of speed, while maintaining comparable performance for solving QSD problems in quantum communication applications.

On Erdős covering systems in global function fields

A covering system of the integers is a finite collection of arithmetic progressions whose union is the set of integers. A well-known problem on covering systems is the minimum modulus problem posed by Erdős in 1950, who asked whether the minimum modulus in such systems with distinct moduli can be arbitrarily large. This problem was resolved by Hough in 2015, who showed that the minimum modulus is at most 1016. In 2022, Balister, Bollobás, Morris, Sahasrabudhe and Tiba reduced Hough's bound to 616,000 by developing Hough's method. They call it the distortion method. In this paper, by applying this method, we mainly prove that there does not exist any covering system of multiplicity s in any global function field of genus g over Fq for q≥(1.14+0.16g)e6.5+0.97gs2. In particular, there is no covering system of Fq[x] with distinct moduli for q≥759.

Multiagent learning for competitive opinion optimization

From a perspective of designing or engineering for opinion formation games in social networks, the opinion maximization (or minimization) problem has been studied mainly for designing seeding algorithms that aim at selecting a subset of nodes to control their opinions. We first define a two-player zero-sum Stackelberg game of competitive opinion optimization by letting the player under study as the leader minimize the sum of expressed opinions by doing so-called “internal opinion design”, knowing that the other adversarial player as the follower is to maximize the same objective by also conducting her own internal opinion design. We furthermore consider multiagent learning, specifically using the Optimistic Gradient Descent Ascent, and analyze its convergence to equilibria in the simultaneous-game version of competitive opinion optimization.

A quadratic optimization program for the inverse elastography problem

In this work we focus on the development of a numerical algorithm for the inverse elastography problem. The goal is to perform an efficient material parameter identification knowing the elastic displacement field induced by a mechanical load. We propose to define the inverse problem through a quadratic optimization program which uses the direct problem formulation to define the objective function. In this way, we end up with a convex minimization problem which attains its minimum at the solution of a linear system. The effectiveness of our method is illustrated through numeral examples.

CellCover Captures Neural Stem Cell Progression in Mammalian Neocortical Development.

Definition of cell classes across the tissues of living organisms is central in the analysis of growing atlases of single-cell RNA sequencing (scRNA-seq) data across biomedicine. Marker genes for cell classes are most often defined by differential expression (DE) methods that serially assess individual genes across landscapes of diverse cells. This serial approach has been extremely useful, but is limited because it ignores possible redundancy or complementarity across genes that can only be captured by analyzing multiple genes simultaneously. We aim to identify discriminating panels of genes. To efficiently explore the vast space of possible marker panels, leverage the large number of cells often sequenced, and overcome zero-inflation in scRNA-seq data, we propose viewing gene panel selection as a variation of the "minimal set-covering problem" in combinatorial optimization. We show that this new method, CellCover, captures cell-class-specific signals in the developing mouse neocortex that are distinct from those defined by DE methods. Transfer learning experiments across mouse, primate, and human data demonstrate that CellCover identifies markers of conserved cell classes in neurogenesis, as well as temporal progression in both progenitors and neurons. Exploring markers of human outer radial glia (oRG, or basal RG) across mammals, we show that transcriptomic elements of this key cell type in the expansion of the human cortex appeared in gliogenic precursors of the rodent before the full program emerged in the primate lineage. We have assembled the public datasets we use in this report at NeMO analytics where the expression of individual genes {NeMO Individual Genes} and marker gene panels can be freely explored {NeMO: Telley 3 Sets Covering Panels}, {NeMO: Telley 12 Sets Covering Panels}, and {NeMO: Sorted Brain Cell Covering Panels}. CellCover is available in {CellCover R} and {CellCover Python}.

Method with batching for stochastic finite-sum variational inequalities in non-Euclidean setting

Variational inequalities are a universal optimization paradigm that incorporate classical minimization and saddle point problems. Nowadays more and more tasks require to consider stochastic formulations of optimization problems. In this paper, we present an analysis of a method that gives optimal convergence estimates for monotone stochastic finite-sum variational inequalities. In contrast to the previous works, our method supports batching, does not lose the oracle complexity optimality and uses an arbitrary Bregman distance to take into account geometry of the problem. Paper provides experimental confirmation to algorithm’s effectiveness.