Unconstrained Nonlinear Optimization Research Articles

A q-Polak\u2013Ribi\xe8re\u2013Polyak conjugate gradient algorithm for unconstrained optimization problems

A Polak–Ribière–Polyak (PRP) algorithm is one of the oldest and popular conjugate gradient algorithms for solving nonlinear unconstrained optimization problems. In this paper, we present a q-variant of the PRP (q-PRP) method for which both the sufficient and conjugacy conditions are satisfied at every iteration. The proposed method is convergent globally with standard Wolfe conditions and strong Wolfe conditions. The numerical results show that the proposed method is promising for a set of given test problems with different starting points. Moreover, the method reduces to the classical PRP method as the parameter q approaches 1.

Application of Spectral PRP Conjugate Gradient Parameter for Unconstrained Optimization Problems

Conjugate Gradient (CG) method have been utilised to solve nonlinear unconstrained optimization problems because of less storage locations and fewer computational cost in dealing with large-scale problems. In this paper, we present a real life application of spectral PRP Conjugate Gradient method in regression analysis, the proposed method is suitably deriving from the CG search direction without secant condition. Some benchmark functions with several variables have been use to prove the global convergence properties and satisfies sufficient descent condition. The numerical results are certifying by exact line search techniques; the method outperform the prominent least square method

A New conjugate gradient method for unconstrained optimization problems with descent property

In this paper, we propose a new conjugate gradient method for solving nonlinear unconstrained optimization. The new method consists of three parts, the first part of them is the parameter of Hestenes-Stiefel (HS). The proposed method is satisfying the descent condition, sufficient descent condition and conjugacy condition. We give some numerical results to show the efficiency of the suggested method.

Optimization of discharge circuit model based on electro pulse boring experiment

High-voltage electro pulse rock-breaking is a green breaking technology, which can achieve high efficiency of rock-breaking. It has a wide application prospect in deep and ultra-deep well drilling and tunnel driving in the future. Pursuant to the time delay characteristics of the gas discharge switch in the electro pulse boring (EPB) system, the equivalent model of the discharge circuit is established based on the time-varying resistance model. With this model, the real-time parameters such as voltage, current and discharge energy can be predicted in the EPB process of different rocks. In line with the arrangement of coaxial cylindrical electrodes, a 25 mm electrode drill bit is designed, and an EPB experiment system is developed based on the transformer step-up pulse power supply. The electro pulse breaking of red sandstone, yellow sandstone, concrete and granite, and the complete drilling of 60 mm diameter in red sandstone and concrete are realized, and EPB rules of different rocks are obtained. The real-time curves of electrical parameters in different rocks are collected by electrical parameter tester. The model parameters are identified based on the least square method of unconstrained nonlinear optimization. The current curve fitted by the identification parameters and the current curve obtained from the experiment are featured in high coincidence and good follow-up, which proves the accuracy of the optimized model and parameter identification. The optimization of the discharge circuit model and the identification of the model parameters based on EPB experiment are of great significance to guide the selection of the technological parameters of EPB, improve the efficiency of rock-breaking and reduce the energy loss in the drilling process.

Hybrid gradient simulated annealing algorithm for finding the global optimal of a nonlinear unconstrained optimization problem

A new hybrid gradient simulated annealing algorithm is introduced. The algorithm is designed to find the global minimizer of a nonlinear function of many variables. The function is assumed to be smooth. The algorithm uses the gradient method together with a line search to ensure convergence from a remote starting point. It is hybridized with a simulated annealing algorithm to ensure convergence to the global minimizer. The performance of the algorithm is demonstrated through extensive numerical experiments on some well-known test problems. Comparisons of the performance of the suggested algorithm and other meta-heuristics methods were reported. It validates the effectiveness of our approach and shows that the suggested algorithm is promising and merits to be implemented in practice.

An alternative tree method for calibration of the local volatility

<p style='text-indent:20px;'>In this paper, we combine the traditional binomial tree and trinomial tree to construct a new alternative tree pricing model, where the local volatility is a deterministic function of time. We then prove the convergence rates of the alternative tree method. The proposed model can price a wide range of derivatives efficiently and accurately. In addition, we research the optimization approach for the calibration of local volatility. The calibration problem can be transformed into a nonlinear unconstrained optimization problem by exterior penalty method. For the optimization problem, we use the quasi-Newton algorithm. Finally, we test our model by numerical examples and options data on the S & P 500 index. Numerical results confirm the excellent performance of the alternative tree pricing model.</p>

A novel high-performance passive non-traditional inerter-based dynamic vibration absorber

This paper presents the performance evaluation for a novel high-performance nontraditional inerter-based dynamic vibration absorber called NIDVA-C4. For the easy physical parameter tuning for the proposed device, the Extended Fixed-Points Technique (EFPT) is applied, and then short closed-form solutions are provided for the undamped systems. Subsequently, performance evaluation is carried out by computing two performance measures, which are the H∞ and H2 performance indices. For both H∞ and H2 criteria, constrained and unconstrained multivariable non-linear optimization problems are formulated, which are solved through the formulation of sets of high-order nonlinear equations. For H∞ optimization, the nondimensional FRF function of NIDVA-C4 is minimized at resonant frequencies considering the harmonic force excitation case. For H2 optimization, two random vibration excitation cases are considered which are the following; random force and ground motion excitation cases. For both random vibration excitation cases, the variances of the squared modulus of the frequency response function are computed for undamped primary structure. Considering the mass ratio range from 1% to 10%, from H∞ performance evaluation, the numerical results revealed that the proposed device yields improvements of 2–15% and 23–33% when compared to the inerter-based dynamic vibration absorber (IDVA-C6) or rotational inertial double tuned mass dampers and classic DVA, respectively. On the other hand, for the random force excitation case, improvements of 1–16% and 14–27% can be obtained. Furthermore, for the random acceleration excitation case, the NIDVA-C4 provides dynamic performances of 2–6% and 15–20% compared with the IDVA-C6 and classic dynamic vibration absorber (DVA), respectively. Finally, to appreciate the vibration attenuation in the time domain, numerical simulations were carried out considering the randomly excited mechanical systems.

Necessary and sufficient optimality conditions for non-linear unconstrained and constrained optimization problem with interval valued objective function

In the theory of non-linear constrained optimization problem, the most familiar Karush Kuhn Tucker (KKT) conditions play an important role. Basically, these conditions are necessary optimality conditions of a constrained optimization problem with equality and inequality constraints. The goal of this paper is to introduce the necessary optimality conditions (named as equivalent KKT conditions) of a constrained optimization problem with interval-valued objective function. Depending on the type of objective function and inequality constraints, all possible cases (i.e., objective function interval-valued and constraints are real-valued, objective function and constraints both are interval-valued, the objective function is real-valued and constraints are interval-valued) have been investigated to find the necessary optimality conditions. For this purpose, this paper deals with the necessary and sufficient conditions of unconstrained optimization problem with interval-valued objective function. Also, with the help of interval order relations the definitions of local and global optima of interval-valued function have been proposed. Finally, in order to illustrate the equivalent KKT conditions of the interval-valued constrained optimization problem and also the necessary as well as sufficient conditions of an unconstrained optimization problem, some numerical examples have been considered and solved.

A method for matching response spectra of endurance time excitations via the Fourier transform

The endurance time (ET) method is a dynamic analysis in which structures are subjected to intensifying excitations, also known as ET excitation functions (ETEF). The ET method is a tool for structural response prediction. The main advantage of the ET method over conventional approaches is its much lower demand for computational efforts. The concept of acceleration spectra is used in generating existing ETEFs. It is expected that ETEF acceleration spectra increase consistently with time and remain proportional to a target spectrum. Nonlinear unconstrained optimization is commonly used to generate ETEFs. Generating new ETEFs is a complicated time-consuming mathematical problem. If the target acceleration spectrum changes, new ETEFs must be generated. This study intends to modify existing ETEFs to be compatible with a desired acceleration spectrum. This process, called spectral matching, obviates the need for using the complicated generating procedure in simulating new ETEFs. ETEFs spectral matching is introduced in this paper for the first time. A Fourier-based method for ETEFs spectral matching is proposed. This algorithm is then applied in a case study. Results are presented to prove the efficiency of the algorithm.

A Lyapunov approach for attraction domain estimation of polynomial discrete nonlinear systems with quadratic and cubic terms

In this paper, we propose a Lyapunov method for estimating asymptotic stability domain of polynomial discrete nonlinear systems, drawn from an existing approach for attraction domain estimation of continuous nonlinear systems. Proceeding from two different forms of a parametric equation, we present two development methods with the aim of identifying a sufficient condition to obtain a part of the stability domain for quadratic and cubic n-dimensional polynomial discrete systems. The results consist in solving a polynomial equation and unconstrained nonlinear optimization problems, where in case of quadratic or cubic systems, significant simplifications are obtained. The main notable advantages of this approach are its applicability on high-dimensional systems and obtaining an ellipsoidal stability domain which approaches the exact attraction domain limits. Two application examples are illustrated to validate each development.

On the convergence of steepest descent methods for multiobjective optimization

In this paper we consider the classical unconstrained nonlinear multiobjective optimization problem. For such a problem, it is particularly interesting to compute as many points as possible in an effort to approximate the so-called Pareto front. Consequently, to solve the problem we define an ???a posteriori??? algorithm whose generic iterate is represented by a set of points rather than by a single one. The proposed algorithm takes advantage of a linesearch with extrapolation along steepest descent directions with respect to (possibly not all of) the objective functions. The sequence of sets of points produced by the algorithm defines a set of ???linked??? sequences of points. We show that each linked sequence admits at least one limit point (not necessarily distinct from those obtained by other sequences) and that every limit point is Pareto-stationary. We also report numerical results on a collection of multiobjective problems that show efficiency of the proposed approach over more classical ones.

Entropy binomial tree method and calibration for the volatility smile

In this paper, we combine the maximum entropy principle with binomial tree to construct a non-recombining entropy binomial tree pricing model under the volatility that is a function of time, and give the rate of convergence. The model may yield an unbiased and objective probability. In addition, we research the calibration problem of volatility with the entropy binomial tree, and adopt an exterior penalty method to transform this problem into a nonlinear unconstrained optimization problem. For the optimization problem, we use the quasi-Newton algorithm. Finally, we test our model by numerical examples and options data on the S&P 500 index. The results confirm the effectiveness of the entropy binomial tree pricing model.

A two-echelon newsvendor problem with limited budget and capacity

In this paper, two-echelon newsvendor problem is considered. Many real-life situations like fashion, food industries, and healthcare services match newsvendor problem. Our problem is determining levels of inventory in order to optimize the profit and service level in selling a product. This product is made up of several raw materials. Only the distribution of demand is known, and the hot season of selling the product is just a short period and after that, the price of the product drops dramatically. The storage space and initial budget are limited. We modeled and solved the problem as an unconstrained nonlinear optimization problem using two nonlinear techniques, the sequential unconstrained minimization technique (SUMT) and steepest descent (SD).

A New Modified Conjugate Gradient Method and Its Global Convergence Theorem

In this article, we try to proposed a new conjugate gradient method for solving unconstrained optimization problems, we focus on conjugate gradient methods applied to the non-linear unconstrained optimization problems, the positive step size is obtained by a line search and the new scalar to the new direction for the conjugate gradient method is derived from the quadratic function and Taylor series and by using quasi newton condition and Newton direction while deriving the new formulae. We also prove that the search direction of the new conjugate gradient method satisfies the sufficient descent and all assumptions of the global convergence property are considered and proved .in order to complete the benefit of our research we should take into account studied the numerical results which are written in FORTRAN language when the objective function is compared our new algorithm with HS and PRP methods on the similar set of unconstrained optimization test problems which is very efficient and encouragement numerical results.

A COMPARATIVE STUDY OF SOME MODIFICATIONS OF CG METHODS UNDER EXACT LINE SEARCH

Conjugate Gradient (CG) method is a technique used in solving nonlinear unconstrained optimization problems. In this paper, we analysed the performance of two modifications and compared the results with the classical conjugate gradient methods of. These proposed methods possesse global convergence properties for general functions using exact line search. Numerical experiments show that the two modifications are more efficient for the test problems compared to classical CG coefficients.

A sufficient descent nonlinear conjugate gradient method for solving [formula omitted]-tensor equations

Tensor equations is a kind of important tensor optimization problems with higher order nonlinear equations, which are widely used in engineering and economics. This paper is concerned with solving M-tensor equations. We transform M-tensor equations to nonlinear unconstrained optimization problems. Then, a sufficient descent nonlinear conjugate gradient method with inexact line search is proposed for solving the transformed unconstrained optimization problem. Under only one mild assumption, the global convergence of the proposed method is proved. Finally, to show the effectiveness of the proposed nonlinear conjugate gradient method, we compare it with three-term conjugate gradient method and Newton method. The numerical results show that the proposed nonlinear conjugate gradient method is potentially efficient.

A New Quasi-Newton (SR1) With PCG Method for Unconstrained Nonlinear Optimization

A New Quasi-Newton (SR1) With PCG Method for Unconstrained Nonlinear Optimization

On the mixed integer randomized pattern search algorithm

We analyze the convergence and performance of a novel direct search algorithm for identifying at least a local minimum of unconstrained mixed integer nonlinear optimization problems. The Mixed Integer Randomized Pattern Search Algorithm (MIRPSA), so-called by the author, is based on a randomized pattern search, which is modified by two main operations for finding at least a local minimum of our problem, namely: moving operation and shrinking operation. The convergence properties of the MIRPSA are here analyzed from a Markov chain viewpoint, which is represented by an infinite countable set of states {d(q)}∞q=0, where each state d(q) is defined by a measure of the qth randomized pattern search Hq, for all q ∈ N. According to the algorithm, when a moving operation is carried out on a qth randomized pattern search Hq, the MIRPSA Markov chain holds its state. Meanwhile, if the MIRPSA carries out a shrinking operation on a qth randomized pattern search Hq, the algorithm will then visit the next (q + 1)th state. Since the MIRPSA Markov chain never goes back to any visited state, we therefore say that the MIRPSA yields a birth and miscarriage Markov chain.

A concise second-order complexity analysis for unconstrained optimization using high-order regularized models

ABSTRACTAn adaptive regularization algorithm is proposed that uses Taylor models of the objective of order p, , of the unconstrained objective function, and that is guaranteed to find a first- and second-order critical point in at most function and derivatives evaluations, where and are prescribed first- and second-order optimality tolerances. This is a simple algorithm and associated analysis compared to the much more general approach in Cartis et al. [Sharp worst-case evaluation complexity bounds for arbitrary-order nonconvex optimization with inexpensive constraints, arXiv:1811.01220, 2018] that addresses the complexity of criticality higher-than two; here, we use standard optimality conditions and practical subproblem solves to show a same-order sharp complexity bound for second-order criticality. Our approach also extends the method in Birgin et al. [Worst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models, Math. Prog. A 163(1) (2017), pp. 359–368] to finding second-order critical points, under the same problem smoothness assumptions as were needed for first-order complexity.

A descent cautious BFGS method for computing US-eigenvalues of symmetric complex tensors

Unitary symmetric eigenvalues (US-eigenvalues) of symmetric complex tensors and unitary eigenvalues (U-eigenvalues) for non-symmetric complex tensors are very important because of their background of quantum entanglement. US-eigenvalue is a generalization of Z-eigenvalue from the real case to the complex case, which is closely related to the best complex rank-one approximations to higher-order tensors. The problem of finding US-eigenpairs can be converted to an unconstrained nonlinear optimization problem with complex variables, their complex conjugate variables and real variables. However, optimization methods often need a first- or second-order derivative of the objective function, and cannot be applied to real valued functions of complex variables because they are not necessarily analytic in their argument. In this paper, we first establish the first-order complex Taylor series and Wirtinger calculus of complex gradient of real-valued functions with complex variables, their complex conjugate variables and real variables. Based on this theory, we propose a norm descent cautious BFGS method for computing US-eigenpairs of a symmetric complex tensor. Under appropriate conditions, global convergence and superlinear convergence of the proposed method are established. As an application, we give a method to compute U-eigenpairs for a non-symmetric complex tensor by finding the US-eigenpairs of its symmetric embedding. The numerical examples are presented to support the theoretical findings.