Nonlinear Semi-infinite Programming Research Articles

Approximate optimality conditions and approximate duality theorems for nonlinear semi-infinite programming problems with uncertainty data

Approximate optimality conditions and approximate duality theorems for nonlinear semi-infinite programming problems with uncertainty data

A new adaptive method to nonlinear semi-infinite programming

<p style='text-indent:20px;'>In this paper, we propose a new adaptive method for solving nonlinear semi-infinite programming(SIP). In the presented method, the continuous infinite inequality constraints are transformed into equivalent equality constraints in integral form. Based on penalty method and trust region strategy, we propose a modified quadratic subproblem, in which an adaptive parameter is considered. The acceptable criterion of the trial point is adjustable according to the value of this adaptive parameter and the improvements that made by the current iteration. Compared with the existing methods, our method is more flexible. Under some reasonable conditions, the convergent properties of the proposed algorithm are proved. The numerical results are reported in the end.</p>

Robust approximate optimal solutions for nonlinear semi-infinite programming with uncertainty

In this paper, we deal with robust approximate quasi optimal solutions for a class of nonlinear semi-infinite programming with data uncertainty in both the objective and constraints. By using a new robust-type subdifferential constraint qualification and some generalized convexity assumptions, we first establish approximate optimality conditions for robust approximate quasi optimal solutions of . Then, we introduce a Mixed-type robust approximate dual problem for and investigate robust approximate duality relations between them. Furthermore, we establish a nonsmooth robust approximate saddle point theorem for an uncertain approximate Lagrangian function associated with .

Filter Trust Region Method for Nonlinear Semi-Infinite Programming Problem

We present a filter trust region method for nonlinear semi-infinite programming. Based on the discretization technique and motivated by the multiobjective programming, we transform the semi-infinite problem into a finite one. Together with the filter technique, we propose a modified method that avoids the merit function. Compared with the existing methods, our method is more flexible and easier to implement. Under some mild conditions, the convergent properties are proved. Moreover, the numerical results are reported in the end.

Distributionally Robust Reward-Risk Ratio Optimization with Moment Constraints

Reward-risk ratio optimization is an important mathematical approach in finance. We revisit the model by considering a situation where an investor does not have complete information on the distribution of the underlying uncertainty and consequently a robust action is taken to mitigate the risk arising from ambiguity of the true distribution. We consider a distributionally robust reward-risk ratio optimization model varied from the ex ante Sharpe ratio where the ambiguity set is constructed through prior moment information and the return function is not necessarily linear. We transform the robust optimization problem into a nonlinear semi-infinite programming problem through standard Lagrange dualization and then use the well-known entropic risk measure to construct an approximation of the semi-infinite constraints. We solve the latter by an implicit Dinkelbach method. Finally, we apply the proposed robust model and numerical scheme to a portfolio optimization problem and report some preliminary numerical ...

A reduction method for nonlinear semi-infinite programming based on an exact penalty technique

Semi-infinite programming (SIP) problems arise in several areas of engineering, such as robot trajectory planning, production planning, design of digital filters and air pollution control. Despite its large applicability there is not much software available to solve such problems. The only available SIP solvers are the fseminf MATLAB function, FSQP and NSIPS, but none of these solvers provide an algorithm belonging to the class of reduction type methods. In this article we propose a reduction type method, based on a penalty technique to solve SIP problems. We report numerical results with 117 test problems from the SIPAMPL database, using a MATLAB implementation of the proposed algorithm, which we coined as SiRedAl.

Embedding Monotonicity in the Construction of Polynomial Open-Circuit Voltage Model for Lithium-Ion Batteries: A Semi-infinite Programming Formulation Approach

Accurate modeling of open-circuit-voltage (OCV) plays important roles both in state-of-charge (SOC) estimation and state-of-health (SOH) monitoring for lithium-ion batteries (LIBs). Monotonicity violation in OCV model would lead to inaccurate SOC estimation and ineffective of incremental capacity analysis (ICA) for SOH monitoring. In this study, first-order derivative of OCV, with respect to SOC is introduced to theoretically ensure the satisfaction of monotonicity and a nonlinear semi-infinite programming (NSIP) problem is constructed to parameter estimation. A global optimization approach via restriction of the right-hand side is used to efficiently and globally optimize the NSIP. Both fitting and ICA results demonstrate the effectiveness of the proposed method. Moreover, in comparison to the traditional polynomial and sigmoid models, the NSIP polynomial model is the best choice for performing further SOC estimation and SOH monitoring. The results thus indicate that a NSIP framework for embedding prior ...

OPTIMALITY CONDITIONS AND DUALITY FOR SEMI-INFINITE PROGRAMMING INVOLVING SEMILOCALLY TYPE I-PREINVEX AND RELATED FUNCTIONS

A nondifferentiable nonlinear semi-infinite programming problem is considered, where the functions involved are <TEX>${\eta}$</TEX>-semidifferentiable type I-preinvex and related functions. Necessary and sufficient optimality conditions are obtained for a nondifferentiable nonlinear semi-in nite programming problem. Also, a Mond-Weir type dual and a general Mond-Weir type dual are formulated for the nondifferentiable semi-infinite programming problem and usual duality results are proved using the concepts of generalized semilocally type I-preinvex and related functions.

Relaxed cutting plane method with convexification for solving nonlinear semi-infinite programming problems

In this paper, we present an algorithm to solve nonlinear semi-infinite programming (NSIP) problems. To deal with the nonlinear constraint, Floudas and Stein (SIAM J. Optim. 18:1187---1208, 2007) suggest an adaptive convexification relaxation to approximate the nonlinear constraint function. The ?BB method, used widely in global optimization, is applied to construct the convexification relaxation. We then combine the idea of the cutting plane method with the convexification relaxation to propose a new algorithm to solve NSIP problems. With some given tolerances, our algorithm terminates in a finite number of iterations and obtains an approximate stationary point of the NSIP problems. In addition, some NSIP application examples are implemented by the method proposed in this paper, such as the proportional-integral-derivative controller design problem and the nonlinear finite impulse response filter design problem. Based on our numerical experience, we demonstrate that our algorithm enhances the computational speed for solving NSIP problems.

A Deterministic Approach for Solving the Hull and White Interest Rate Model

This work considers the resolution of the Hull and White interest rate model. A deterministic process is adopted to model the random behavior of interest rate variation as a deterministic perturbation. It shows that the interest rate function and the yield function of the Hull and White interest rate model can be obtained by solving a nonlinear semi-infinite programming problem. A relaxed cutting plane algorithm is then proposed for the resulting optimization problem. The features of the proposed method are tested using a set of real data and compared with some commonly used spline fitting methods.

A relaxed cutting plane algorithm for solving the Vasicek-type forward interest rate model

This work considers the solution of the Vasicek-type forward interest rate model. A deterministic process is adopted to model the random behavior of interest rate variation as a deterministic perturbation. It shows that the solution of the Vasicek-type forward interest rate model can be obtained by solving a nonlinear semi-infinite programming problem. A relaxed cutting plane algorithm is then proposed for solving the resulting optimization problem. The features of the proposed method are tested using a set of real data and compared with some commonly used spline fitting methods.

On the resolution of the Vasicek-type interest rate model

This work considers the resolution of the Vasicek-type interest rate model. A deterministic process is adopted to model the random behaviour of interest rate variation as a deterministic perturbation. It shows that the solution of the Vasicek-type interest rate model can be obtained by solving a non-linear semi-infinite programming problem. A relaxed cutting plane algorithm is then proposed for solving the resulting optimization problem. The numerical results illustrate that our approach essentially generates the yield functions with minimal fitting errors and small oscillation.

Optimality conditions and duality for semi-infinite programming involving B-arcwise connected functions

In this paper, a class of functions called B-arcwise connected (BCN) and strictly B-arcwise connected (STBCN) functions are introduced by relaxing definitions of arcwise connected function (CN) and B-vex function. The differential properties of B-arcwise connected function (BCN) are studied. Their two extreme properties are proved. The necessary and sufficient optimality conditions are obtained for the nondifferentiable nonlinear semi-infinite programming involving B-arcwise connected (BCN) and strictly B-arcwise connected (STBCN) functions. Mond-Weir type duality results have also been established.

A reduction type penalty algorithm for nonlinear semi-infinite programming

Semi-infinite programming (SIP) problems arise in several engineering areas such as, for example, robotic trajectory planning, production planning, digital filter design and air pollution control. In spite of being an active research area with many seminal works it lacks available software that could be used by the research community. The only exceptions are the fseminf MATLAB function, available in the Optimization Toolbox, and the NSIPS solver, but neither of them provide an implementation of a method belonging to the well known reduction type class.This paper proposes an implementation of a reduction type algorithm base on a penalty technique and provides a compare between several well known penalty functions.The provided numerical results with a significant number of SIP test problems are reported as performance profiles.

Necessary conditions and duality for inexact nonlinear semi-infinite programming problems

First order necessary conditions and duality results for general inexact nonlinear programming problems formulated in nonreflexive spaces are obtained. The Dubovitskii–Milyutin approach is the main tool used. Particular cases of linear and convex programs are also analyzed and some comments about a comparison of the obtained results with those existing in the literature are given.

A quasi-Newton interior point method for semi-infinite programming

We propose an interior point method for solving nonlinear semi-infinite programming (SIP) problems. The method is based on a transcription of the SIP problem into a nonlinear finite problem. We apply the interior point strategy to the finite problem and obtain an algorithm for nonlinear SIP. The evaluation of the constraint functions and derivatives of the finite problem requires the numerical evaluation of integrals that we compute by an adaptative trapezoid formula. Numerical results with a collection of several problems from the SIPAMPL [A.I.F. Vaz, E.M.G.P. Fernandes and M.P.S.F. Gomes. SIPAMPL v2.0: Semi-infinite programming with AMPL. Technical Report ALG/EF/4-2002, Universidade do Minho, Braga, Portugal, December 2002. [http://www.norg.uminho.pt/aivaz/].] database are also shown. †E-mail: emgpf@dps.uminho.pt ‡E-mail: p.gomes@ic.ac.uk

A sequential quadratic programming with a dual parametrization approach to nonlinear semi-infinite programming

In the sequel of the work reported in Liu et al. (1999), in which a method based on a dual parametrization is used to solve linear-quadratic semi-infinite programming (SIP) problems, a sequential quadratic programming technique is proposed to solve nonlinear SIP problems. A merit function to measure progress toward the solution and a procedure to compute the penalty parameter are also proposed.

Numerical treatment of an asset price model with non-stochastic uncertainty

In contrast to stochastic differential equation models used for the calculation of the term structure of interest rates, we develop an approach based on linear dynamical systems under non-stochastic uncertainty with perturbations. The uncertainty is described in terms of known feasible sets of varying parameters. Observations are used in order to estimate these parameters by minimizing the maximum of the absolute value of measurement errors, which leads to a linear or nonlinear semi-infinite programming problem. A regularized logarithmic barrier method for solving (ill-posed) convex semi-infinite programming problems is suggested. In this method a multi-step proximal regularization is coupled with an adaptive discretization strategy in the framework of an interior point approach. A special deleting rule permits one to use only a part of the constraints of the discretized problems. Convergence of the method and its stability with respect to data perturbations in the cone of convexC 1-functions are studied. On the basis of the solutions of the semi-infinite programming problems a technical trading system for future contracts of the German DAX is suggested and developed.

The Direct Solution of Nonconvex Nonlinear FIR Filter Design Problems by a SIP Method

FIR filter design problems in the frequency domain are nonlinear (semi-infinite) optimization problems. In practice, however, these almost always have not been approached directly, but been solved in a simplified form and/or only under restricting assumptions. In this paper, quite general mathematical formulations of the four main design approximation problems in the frequency domain are presented, which enable the derivation of theoretical results (collected here from R. Reemtsen, 2000b, 2000c) and the application of general-purpose optimization procedures to their direct solution. For the actual solution, a nonlinear semi-infinite programming method from the thesis (S. Gorner, Ph.D. Thesis, Technische Universitat Berlin, Berlin, 1997) of the first author is discussed and applied to several specific design problems. In some cases, the computed solution of the nonlinear problem is compared with that of a convex approximation of the problem.

A semi-infinite programming model for earliness/tardiness production planning with simulated annealing

To describe the earliness/tardiness production planning problems in the JIT environment, a nonlinear semi-infinite programming model was proposed in [1]. Due to a nonconvex objective function and infinitely many constraints, the model is difficult to solve by traditional methods. In this paper, simulated annealing method combined with a heuristic and the steepest descent method is developed. Numerical results shows that the present approach is very efficient. In theory, the developed method is an attempt to solve a continuous domain problem by simulated annealing.