Unconstrained Nonlinear Programming Research Articles

Observer-based reliable stabilization of uncertain linear systems subject to actuator faults, saturation, and bounded system disturbances

A matrix inequality approach is proposed to reliably stabilize a class of uncertain linear systems subject to actuator faults, saturation, and bounded system disturbances. The system states are assumed immeasurable, and a classical observer is incorporated for observation to enable state-based feedback control. Both the stability and stabilization of the closed-loop system are discussed and the closed-loop domain of attraction is estimated by an ellipsoidal invariant set. The resultant stabilization conditions in the form of matrix inequalities enable simultaneous optimization of both the observer gain and the feedback controller gain, which is realized by converting the non-convex optimization problem to an unconstrained nonlinear programming problem. The effectiveness of proposed design techniques is demonstrated through a linearized model of F-18 HARV around an operating point.

A Generalized Computer Technique for Solving Unconstrained Non-Linear Programming Problems

An unconstrained problem with nonlinear objective function has many applications. This is often viewed as a discipline in and of itself. In this paper, we develop a computer technique for solving nonlinear unconstrained problems in a single framework incorporating with Golden section, Gradient Search method. For this, we first combine this algorithm and then develop a generalized computer technique using the programming language MATHEMATICA. We demonstrate our computer technique with a number of numerical examples. Dhaka Univ. J. Sci. 61(1): 75-80, 2013 (January) DOI: http://dx.doi.org/10.3329/dujs.v61i1.15100

An Improved Control Vector Iteration Approach for Nonlinear Dynamic Optimization (I) Problems Without Path Constraints

This study proposes an efficient indirect approach for general nonlinear dynamic optimization problems without path constraints. The approach incorporates the virtues both from indirect and direct methods: it solves the optimality conditions like the traditional indirect methods do, but uses a discretization technique inspired from direct methods. Compared with other indirect approaches, the proposed approach has two main advantages: (1) the discretized optimization problem only employs unconstrained nonlinear programming (NLP) algorithms such as BFGS (Broyden-Fletcher-Goldfarb-Shanno), rather than constrained NLP algorithms, therefore the computational efficiency is increased; (2) the relationship between the number of the discretized time intervals and the integration error of the four-step Adams predictor-corrector algorithm is established, thus the minimal number of time intervals that under desired integration tolerance can be estimated. The classic batch reactor problem is tested and compared in detail with literature reports, and the results reveal the effectiveness of the proposed approach. Dealing with path constraints requires extra techniques, and will be studied in the second paper.

Best proximity point theorems: exposition of a significant non-linear programming problem

The primary goal of this work is to address the non-linear programming problem of globally minimizing the real valued function x ? d(x, Tx) where T is presumed to be a non-self mapping that is a generalized proximal contraction in the setting of a metric space. Indeed, an iterative algorithm is presented to determine a solution of the preceding non-linear programming problem that focuses on global optimization. As a sequel, one can compute optimal approximate solutions to some fixed point equations and optimal solutions to some unconstrained non-linear programming problems.

Stochastic traffic assignment, Lagrangian dual, and unconstrained convex optimization

In this paper, traffic assignment problems with stochastic travel cost perceptions are reformulated and investigated in a new unconstrained nonlinear programming formulation. The objective function of the unconstrained formulation consists of two terms, in which the first term specifies the routing principle of the target problem through a satisfaction function and the sum of the first and second terms denotes the system cost or optimization objective. This formulation proves to be the Lagrangian dual of a generic primal formulation proposed by Maher et al. (2005) for the stochastic system-optimal problem. The primal–dual modeling framework presents such a common functional form that can accommodate a wide range of different traffic assignment problems. Our particular attention is given to the dual formulation in that its unconstrained feature opens the door of applying unconstrained optimization algorithms for its embraced traffic assignment problems. Numerical examples are provided to support the insights and facts derived from applying the primal and dual formulations to model stochastic system-optimal and user-equilibrium problems and justify the conjugate relationship between the primal and dual models.

Study of a constructed function for approximating mode field in photonic crystal fibers

A function is constructed to approximate the fundamental mode field precisely in photonic crystal fibers (PCFs) using least square error criteria. Through the unconstrained nonlinear programming method, the optimum parameters of the constructed function are derived, and the optimum constructed function is found. For photonic crystal fibers, using such an optimum constructed function, small errors are brought out when approximating their fundamental mode fields regardless of d/Λ values. Furthermore, the constructed function is not only suitable for index-guiding PCFs, but also for bandgap PCFs with hollow core to some extent. Based on this constructed function, an analytic expression for the far field of the approximated fundamental mode field is deduced. Numerical results demonstrate that the analytic expression brings out small errors compared with the actual far field.

COMBINATION ADAPTIVE TRUST REGION METHOD BY NON-MONOTONE STRATEGY FOR UNCONSTRAINED NONLINEAR PROGRAMMING

In this paper, we present a new trust region method for unconstrained nonlinear programming in which we blend adaptive trust region algorithm by non-monotone strategy to propose a new non-monotone trust region algorithm with automatically adjusted radius. Both non-monotone strategy and adaptive technique can help us introduce a new algorithm that reduces the number of iterations and function evaluations. The new algorithm preserves the global convergence and has local superlinear and quadratic convergence under suitable conditions. Numerical experiments exhibit that the new trust region algorithm is very efficient and robust for unconstrained optimization problems.

Scale Efficiency Measurement in Data Envelopment Analysis with Interval Data: A Two-Level Programming Approach

Conventional data envelopment analysis (DEA) for measuring the relative efficiency of a set of decisionmaking units (DMUs) requires the observations to have precise values. When observations are imprecise and represented by interval values, the efficiencies are also expected to reflect interval values. Several methods exist to calculate the interval overall and technical efficiencies, but such methods are unable to calculate the interval scale efficiency. The focus of this paper is the application of a two-level programming technique to formulate the problem of determining the bounds of the interval scale efficiency. The associated models are essentially nonlinear programs with only bound constraints for variables in a sophisticated form. Hence, one can modify the conventional quasi-Newton method for unconstrained nonlinear programming problems to solve the two-level programs. Two examples with interval data, one hypothetical and one real, aid in explaining the proposed method and the properties of the results.

Extended local search algorithm based on nonlinear programming for two-dimensional irregular strip packing problem

This paper presents an extended local search algorithm (ELS) for the irregular strip packing problem. It adopts two neighborhoods, swapping two given polygons in a placement and placing one polygon into a new position. The local search algorithm is used to minimize the overlap on the basis of the neighborhoods mentioned above and the unconstrained nonlinear programming model is adopted to further minimize the overlap during the search process. Moreover, the tabu search algorithm is used to avoid local minima, and a compact algorithm is presented to improve the result. The results of standard test instances indicate that when compared with other existing algorithms, the presented algorithm does not only show some signs of competitive power but also updates several best known results.

A dynamic convexized method for nonconvex mixed integer nonlinear programming

We consider in this paper the nonconvex mixed-integer nonlinear programming problem. We present a mixed local search method to find a local minimizer of an unconstrained nonconvex mixed-integer nonlinear programming problem. Then an auxiliary function which has the same global minimizers and the same global minimal value as the original problem is constructed. Minimization of the auxiliary function using our local search method can escape successfully from previously converged local minimizers by taking increasing values of parameters. For the constrained nonconvex mixed-integer nonlinear programming problem, we develop a penalty based method to convert the problem into an unconstrained one, and then use the above method to solve the later problem. Numerical experiments and comparisons on a set of MINLP benchmark problems show the effectiveness of the proposed algorithm.

Nonlinear Predictive Control of Fed‐Batch Cultures of Escherichia coli

AbstractA strategy for controlling a fed‐batch Escherichia coli culture is described to maintain the culture at the boundary between oxidative and oxido‐fermentative regimes. A nonlinear predictive controller is designed to regulate the acetate concentration, constraining the feed rate to follow an optimal reference profile which maximizes the biomass growth. For the sake of simplicity and efficiency, the original problem is converted into an unconstrained nonlinear programming problem, solved by control vector parameterization techniques. The robustness of the structure is further improved by explicitly including the difference between system and model prediction. A robustness study based on a Monte Carlo approach is used to evaluate the performance of the proposed controller. This control law is finally compared to the generic model control strategy.

On Clamping Planning in Workpiece-Fixture Systems

Deformations of contacts between the workpiece and locators/clamps resulting from large contact forces cause overall workpiece displacement, and affect the localization accuracy of the workpiece. An important characteristic of a workpiece-flxture system is that locators are passive elements and can only react to clamping forces and external loads, whereas clamps are active elements and apply a predetermined normal load to the surface of workpiece to prevent it from losing contact with the locators. Clamping forces play an important role in determining the final workpiece quality. This paper presents a general method for determining the optimal clamping forces including their magnitudes and positions. First, we derive a set of compatibility equations that describe the relationship between the displacement of the workpiece and the deformations at contacts. Further, we develop a locally elastic contact model to characterize the nonlinear coupling between the contact force and elastic deformation at the individual contact. We define the minimum norm of the elastic deformations at contacts as the objective function, then formulate the problem of determining the optimal clamping forces as a constrained nonlinear programming problem which guarantees that the fixturing of the workpiece is force closure. Using the exterior penalty function method, we transform the constrained nonlinear programming into an unconstrained nonlinear programming which is, in fact, the nonlinear least square. Consequently, the optimal magnitudes and positions of clamping forces are obtained by using the Levenberg-Marquardt method which is globally convergent. The proposed planning method of optimal clamping forces, which may also have an application to other passive, indeterminate problems such as power grasps in robotics, is illustrated with numerical example.

A new class of projection and contraction methods for solving variational inequality problems

This paper presents a new class of projection and contraction methods for solving monotone variational inequality problems. The methods can be viewed as combinations of some existing projection and contraction methods and the method of shortest residuals, a special case of conjugate gradient methods for solving unconstrained nonlinear programming problems. Under mild assumptions, we show the global convergence of the methods. Some preliminary computational results are reported to show the efficiency of the methods.

Finite Word Length Implementation for Digital Reduced Order Observer Based Controllers

AbstractImplementation issues for digital reduced‐order observer‐based controllers with Finite Word Length (FWL) considerations is studied. A tractable FWL stability related measure is derived, and the optimal FWL realization problem for digital reduced‐order obverser‐based controller is to find those realizations that maximize this related measure. This optimization problem is formulated as an unconstrained nonlinear programming problem which can be solved using the simplex search algorithm. A numerical example is given to illustrate the design procedure and the effectiveness of the proposed method.

Solving Variational Inequalities in a Fuzzy Environment

This paper deals with the variational inequalities in a fuzzy environment. It shows that solving the fuzzy variational inequalities is equivalent to solving a fuzzy generalized complementarity problem. By using the tolerance approach, we show that the fuzzy generalized complementarity problem can be converted to a regular unconstrained nonlinear programming problem. An entropic regularization technique is then applied to solve the resulting optimization problem. Some computational results are included.

Optimal development of doubly curved surfaces

This paper presents algorithms for optimal development (flattening) of a smooth continuous curved surface embedded in three-dimensional space into a planar shape. The development process is modeled by in-plane strain (stretching) from the curved surface to its planar development. The distribution of the appropriate minimum strain field is obtained by solving a constrained nonlinear programming problem. Based on the strain distribution and the coefficients of the first fundamental form of the curved surface, another unconstrained nonlinear programming problem is solved to obtain the optimal developed planar shape. The convergence and complexity properties of our algorithms are analyzed theoretically and numerically. Examples show the effectiveness of the algorithms.

Unconstrained optimisation for the design of distillation columns

In the traditional constrained formulation the separation objectives are posed as inequality constraints. The optimisation variables are the number of stages on each section of the column, the reflux ratio and the product flow rates. The main objective of this work is to reformulate the original constrained problem into an unconstrained problem. The reflux ratio and product flow rate are calculated to satisfy the two active constraints at each stage of the optimisation problem, adding two equations to the modelling system of the distillation column. The design problem of a distillation column is formulated as an unconstrained optimisation problem where the number of stages in each section of the column is treated as continuous optimisation variable. Therefore an unconstrained non-linear programming problem is solved. The objective function used corresponds to the minimisation of the total cost, thus leading to a convex problem. A numerical example corresponding to a real column is included. The total cost versus the number of stages in each section of the column is graphically shown.

Trust region method in neural network

A Hopfield-type neural network with adaptively changing synaptic weights and activation function parameters is presented to solve unconstrained nonlinear programming problems. The network performance is similar to that of the trust region method in the mathematical programming literature. There is a sub-network to solve quadratic programming problems with simple upper and lower bounds. By sequentially activating the sub-network under the control of an externul computer or a special analog or digital processor that adjusts the weights and parameters, the network solves a sequence of unconstrained nonlinear programming problems. Convergence proof and numerical results are given.

Structural optimization using unconstrained nonlinear goal programming algorithm

This paper presents a method for solving structural optimization problems using nonlinear goal programming techniques. The developed method removes the difficulty of having to define an objective function and constraints. It also has the capacity of handling rank ordered design objectives or goals. The formulation of the structural optimization problem into a goal programming form is discussed. The resulting optimization problem is solved using Powell's unconstrained conjugate direction search algorithm. To demonstrate the effectiveness of the method, as a design tool, the solutions of some numerical test cases are included.

A trajectory-following method for unconstrained optimization

A trajectory-following method with interesting properties is considered for solving unconstrained nonlinear programming problems. The trajectory is defined by a special system of ordinary differential equations. This system uses only the gradient of the objective function. Numerical examples are given.