Filled functions for unconstrained global optimization

This paper is concerned with filled function techniques for unconstrained global minimization of a continuous function of several variables. More general forms of filled functions are presented for smooth and non-smooth optimization problems. These functions have either one or two adjustable parameters. Conditions on functions and on the values of parameters are given so that the constructed functions have the desired properties of filled functions.     

Melnikov functions and limit cycles in piecewise smooth perturbations of a linear center using regularization method

In this article we study limit cycles in piecewise smooth perturbations of a linear center. In this setting it is common to adapt classical results for smooth systems, like Melnikov functions, to non-smooth ones. However, there is little justification for this procedure in the literature. By using the regularization method we give a theoretical proof that supports the use of Melnikov functions directly from the original non-smooth problem.     

Variable Programming: A Generalized Minimax Problem. Part II: Algorithms

In this part of the two-part series of papers, algorithms for solving some variable programming (VP) problems proposed in Part I are investigated. It is demonstrated that the non-differentiability and the discontinuity of the maximum objective function, as well as the summation objective function in the VP problems constitute difficulty in finding their solutions. Based on the principle of statistical mechanics, we derive smooth functions to approximate these non-smooth objective functions with specific activated feasible sets. By transforming the minimax problem and the corresponding variable programming problems into their smooth versions we can solve the resulting problems by some efficient algorithms for smooth functions. Relevant theoretical underpinnings about the smoothing techniques are established. The algorithms, in which the minimization of the smooth functions is carried out by the standard quasi-Newton method with BFGS formula, are tested on some standard minimax and variable programming problems. The numerical results show that the smoothing techniques yield accurate optimal solutions and that the algorithms proposed are feasible and efficient.This work was supported by the RGC grant CUHK 152/96H of the Hong Kong Research Grant Council.     

Characteristics of semi-convex frontier optimization

We study semi-convex frontier (SCF) optimization problems where objective functions can be semi-convex and constraint sets can be non-polyhedron, which stem from a growing range of optimization applications such as frontier analysis, multi-objective programming in economics. The new findings of this paper can be summarized as follows: (1) We characterize non-dominated points of a non-polyhedron optimal solution set of a semi-convex frontier program. (2) We obtain optimality conditions of a constant modulus SCF program, of which the objective function is semi-convex with a constant semiconvexity modulus. (3) We obtain a non-smooth Hölder stability of the optimal solutions of a semiconvex frontier program. (4) We use generalized differentiability to establish sensitivity analysis of the optimal value function of a semi-convex frontier program.     

Clifford algebra valued boundary integral equations for three-dimensional elasticity

Applications of Clifford analysis to three-dimensional elasticity are addressed in the present paper. The governing equation for the displacement field is formulated in terms of the Dirac operator and Clifford algebra valued functions so that a general solution is obtained analytically in terms of one monogenic function and one multiple-component spatial harmonic function together with its derivative. In order to solve numerically the three-dimensional problems of elasticity for an arbitrary domain with complicated boundary conditions, Clifford algebra valued boundary integral equations (BIEs) for multiple-component spatial harmonic functions at an observation point, either inside the domain, on the boundary, or outside the domain, are constructed. Both smooth and non-smooth boundaries are considered in the construction. Moreover, the singularities of the integrals are evaluated exactly so that in the end singularity-free BIEs for the observation point on the boundary taking values on Clifford numbers can be obtained. A Clifford algebra valued boundary element method (BEM) based on the singularity-free BIEs is then developed for solving three-dimensional problems of elasticity. The accuracy of the Clifford algebra valued BEM is demonstrated numerically.     

A combined constraint handling framework: an empirical study

This paper presents a new combined constraint handling framework (CCHF) for solving constrained optimization problems (COPs). The framework combines promising aspects of different constraint handling techniques (CHTs) in different situations with consideration of problem characteristics. In order to realize the framework, the features of two popular used CHTs (i.e., Deb’s feasibility-based rule and multi-objective optimization technique) are firstly studied based on their relationship with penalty function method. And then, a general relationship between problem characteristics and CHTs in different situations (i.e., infeasible situation, semi-feasible situation, and feasible situation) is empirically obtained. Finally, CCHF is proposed based on the corresponding relationship. Also, for the first time, this paper demonstrates that multi-objective optimization technique essentially can be expressed in the form of penalty function method. As CCHF combines promising aspects of different CHTs, it shows good performance on the 22 well-known benchmark test functions. In general, it is comparable to the other four differential evolution-based approaches and five dynamic or ensemble state-of-the-art approaches for constrained optimization.     

Derivatives and subderivatives of buffered probability of exceedance

In this letter, we study the derivatives and subderivatives of buffered probability of exceedance (bPOE), in which we provide the mathematical expressions with rigorous proofs for the case when bPOE is smooth. Furthermore, we extend the study to a general non-smooth case for which a set of quasigradients are explored, under a mild assumption, i.e., the corresponding random function with respect to the decision variable is convex.     

On the Global Convergence of a Projective Trust Region Algorithm for Nonlinear Equality Constrained Optimization

A trust-region sequential quadratic programming (SQP) method is developed and analyzed for the solution of smooth equality constrained optimization problems. The trust-region SQP algorithm is based on filter line search technique and a composite-step approach, which decomposes the overall step as sum of a vertical step and a horizontal step. The algorithm includes critical modifications of horizontal step computation. One orthogonal projective matrix of the Jacobian of constraint functions is employed in trust-region subproblems. The orthogonal projection gives the null space of the transposition of the Jacobian of the constraint function. Theoretical analysis shows that the new algorithm retains the global convergence to the first-order critical points under rather general conditions. The preliminary numerical results are reported.     

Non-linear second-order periodic systems with non-smooth potential

In this paper we study second order non-linear periodic systems driven by the ordinary vectorp-Laplacian with a non-smooth, locally Lipschitz potential function. Our approach is variational and it is based on the non-smooth critical point theory. We prove existence and multiplicity results under general growth conditions on the potential function. Then we establish the existence of non-trivial homoclinic (to zero) solutions. Our theorem appears to be the first such result (even for smooth problems) for systems monitored by thep-Laplacian. In the last section of the paper we examine the scalar non-linear and semilinear problem. Our approach uses a generalized Landesman-Lazer type condition which generalizes previous ones used in the literature. Also for the semilinear case the problem is at resonance at any eigenvalue.     

A STOCHASTIC MOVING BALLS APPROXIMATION METHOD OVER A SMOOTH INEQUALITY CONSTRAINT

We consider the problem of minimizing the average of a large number of smooth component functions over one smooth inequality constraint. We propose and analyze a stochastic Moving Balls Approximation (SMBA) method. Like stochastic gradient (SG) methods, the SMBA method's iteration cost is independent of the number of component functions and by exploiting the smoothness of the constraint function, our method can be easily implemented. Theoretical and computational properties of SMBA are studied, and convergence results are established. Numerical experiments indicate that our algorithm dramatically outperforms the existing Moving Balls Approximation algorithm (MBA) for the structure of our problem.     

Bayesian isotonic changepoint analysis

A general approach to Bayesian isotonic changepoint problems is developed. Such isotonic changepoint analysis includes trends and other constraint problems and it captures linear, non-smooth as well as abrupt changes. Desired marginal posterior densities are obtained using a Markov chain Monte Carlo method. The methodology is exemplified using one simulated and two real data examples, where it is shown that our proposed Bayesian approach captures the qualitative conclusion about the shape of the trend change.     

Fractal Approximation

In the present article every complex square integrable function defined in a real bounded interval is approached by means of a complex fractal function. The approximation depends on a partition of the interval and a vectorial parameter of the iterated function system providing the fractal attractor. The original may be discontinuous or undefined in a set of zero measure. The fractal elements can modify the features of the originals, for instance their character of smooth or non-smooth. The properties of the operator mapping every function into its fractal analogue are studied in the context of the uniform and least square norms. In particular, the transformation provides a decomposition of the set of square integrable maps. An orthogonal system of fractal functions is constructed explicitly for this space. Sufficient conditions for the uniform convergence of the fractal series expansion corresponding to this basis are also deduced. The fractal approximation of real functions is obtained as a particular case.     

非对称碰撞约束下形状记忆合金梁的随机响应与分岔

形状记忆合金(SMA)是二十一世纪具有形状记忆效应的新型智能材料.针对具有非对称约束的SMA梁,本文构造了碰撞振动系统.在无碰撞和有碰撞两种情况下,利用随机平均法给出了近似解析结果.数值模拟作为验证解析结果的工具.结果表明,系统能量的概率响应曲线具有非光滑特性.当约束位置发生变化时,系统会出现随机P分岔和D分岔.     

Chaotic attractor generation and critical value analysis via switching approach

The generation of novel chaotic funnel-shaped attractors is introduced and the analysis of related critical values is given with a proposed switching method in this paper. The underlying mechanism involves a simple three-dimensional switched system and a hysteretically switching signal. Moreover, theoretic analysis is carried out to study the attractor generation and the corresponding critical values by fully utilizing the specific structure of the non-smooth system. Based on carefully derivation, the critical values and related stability regions of the created attractors are estimated explicitly, which is usually impossible for general non-smooth dynamics. In addition, it is demonstrated by simulation that various attractor patterns are generated conveniently by adjusting suitable system parameters.     

Differential stability in non-Lipschitzian optimization

In this paper, upper and lower bounds are established for the Dini directional derivatives of the marginal function of an inequality-constrained mathematical program with right-hand-side perturbations. A nonsmooth analogue of the Cottle constraint qualification is assumed, but the objective and constraint functions are not assumed to be differentiable, convex, or locally Lipschitzian. Our upper bound sharpens previous results from the locally Lipschitzian case by means of a subgradient smaller than the Clarke generalized gradient. Examples demonstrate, however, that a corresponding strengthening of the lower bound is not possible. Corollaries of this work include general criteria for exactness of penalty functions as well as information on the relationship between calmness and other constraint qualifications in nonsmooth optimization.The author is grateful for the helpful comments of a referee.     

一类非光滑凸优化问题的邻近梯度算法

考虑求解目标函数为光滑损失函数与非光滑正则函数之和的凸优化问题的一种基于线搜索的邻近梯度算法及其收敛性分析,证明了在梯度局部Lipschitz连续条件下该算法是R-线性收敛的,并在非光滑部分为稀疏块LASSO正则函数情况下给出了误差界条件成立的证明,得到了线性收敛率.最后,数值实验结果验证了方法的有效性.     

Constrained Nonconvex Nonsmooth Optimization via Proximal Bundle Method

In this paper, we consider a constrained nonconvex nonsmooth optimization, in which both objective and constraint functions may not be convex or smooth. With the help of the penalty function, we transform the problem into an unconstrained one and design an algorithm in proximal bundle method in which local convexification of the penalty function is utilized to deal with it. We show that, if adding a special constraint qualification, the penalty function can be an exact one, and the sequence generated by our algorithm converges to the KKT points of the problem under a moderate assumption. Finally, some illustrative examples are given to show the good performance of our algorithm.     

Metric Sample Spaces of Continuous Geometric Curves and Estimation of Their Centroids

The metric sample space of Fréechet curves (Fréechet , 1934,1951,1961) is based on a generalization of regular curves that covers continuous curves in full generality. This makes it possible to deal with both smooth and non-smooth, even non-rectifiable geometric curves in statistical analysis. In the present paper this sample space is further extended in two directions that are relevant in practice: to incorporate information on landmark points in the curves and to impose invariance with respect to an arbitrary group of isometric spatial transformations. Properties of the introduced sample spaces of curves are studied, specially those concerning to the generation and representation of random curves by random functions.In order to provide measures of central tendency and dispersion of random curves, centroids and restricted centroids ofrandom curves are defined in a general metric framework, and methods for their consistent estimation are derived.     

Efficiently solving total least squares with Tikhonov identical regularization

The Tikhonov identical regularized total least squares (TI) is to deal with the ill-conditioned system of linear equations where the data are contaminated by noise. A standard approach for (TI) is to reformulate it as a problem of finding a zero point of some decreasing concave non-smooth univariate function such that the classical bisection search and Dinkelbach’s method can be applied. In this paper, by exploring the hidden convexity of (TI), we reformulate it as a new problem of finding a zero point of a strictly decreasing, smooth and concave univariate function. This allows us to apply the classical Newton’s method to the reformulated problem, which converges globally to the unique root with an asymptotic quadratic convergence rate. Moreover, in every iteration of Newton’s method, no optimization subproblem such as the extended trust-region subproblem is needed to evaluate the new univariate function value as it has an explicit expression. Promising numerical results based on the new algorithm are reported.     

D’Alembert function for exact non-smooth modal analysis of the bar in unilateral contact

Non-smooth modal analysis is an extension of modal analysis to non-smooth systems, prone to unilateral contact conditions for instance. The problem of a one-dimensional bar subject to unilateral contact on its boundary has been previously investigated numerically and the corresponding spectrum of vibration could be partially explored. In the present work, the non-smooth modal analysis of the above system is reformulated as a set of functional equations through the use of both d’Alembert solution to the wave equation and the method of steps for Neutral Delay Differential Equations. The system features a strong internal resonance condition and it is established that irrational and rational periods of vibration should be carefully distinguished. For irrational periods, it was previously proven that the displacement field of the non-smooth modes of vibration is characterized with piecewise-linear functions in space and time and such a motion is unique for a prescribed energy. However, for rational periods, which are the subject of this work, new periodic solutions are found analytically. Findings consist of families of iso-periodic solutions with piecewise-smooth displacement fields in space and time and continua of piecewise-smooth periodic solutions of the same energy and frequency.