Method Of Feasible Directions Research Articles

Study of Methods of Nonlinear Optimization Using Computer Means

In this paper we consider a program-methodical system for learning nonlinear optimization methods. The FIRST PART of the system is the theory of constructing and investigations of methods for solving nonlinear extremal problems. It is based on the usage of linearization of „active“ constraints and „reduced“ direction. Under the united scheme algorithms of the first and second orders of widely known groups of methods are constructed: methods of the nondifferentiable cost functions, the differentiable cost and barrier functions, methods of the modified Lagrangian functions, methods of centers, methods of feasible directions. The optimization dialog system ODiS for Windows′95/NT for solving nonlinear optimization problems is the SECOND PART of the system. Algorithms mentioned above are realized in the system under the united scheme. The bank of test problems which illustrates the usage of investigated methods is the THIRD PART of the system.

Optimum design of metallic corrugated core sandwich panels subjected to blast loads

Recently, the naval industry has continued to develop innovative lightweight structural concepts with the purpose of seeking alternative replacements for conventional plate–beam metallic structures in selected areas of ships. This study investigated the optimum design of metallic corrugated core sandwich panels subjected to blast loads by using a combined algorithm, the Feasible Direction Method (FDM) coupled with the Backtrack Program Method (BPM), in which the corrugation leg, corrugation angle, face sheet thickness, core thickness and corrugation pitch are selected as design variables, and the axial compression, bending and buckling constraints and the side constraints of the manufacturing limitations on the sizes are considered. A corrugated core sandwich panel model in the combatant deckhouse of a naval ship is adopted in the optimum study. The results show that the corrugation leg, corrugation angle and core thickness are most important for the core component, and that the corrugation leg and face sheet thickness are most important for the face sheet component. The significance of the design varibles is discussed in detail. Optimization results may provide a useful reference for designers.

Influence from numerical noise in the objective function for flow design optimisation

The overall pressure drop in an axisymmetric contraction is minimised using two different grid sizes. The transition region was parameterised with only two design variables to make it possible to create surface plots of the objective function in the design space, which were based on 121 CFD calculations for each grid. The coarse grid showed to have significant numerical noise in the objective function while the finer grid had less numerical noise. The optimisation was performed with two methods, a Response Surface Model (RSM) and a gradient‐based method (the Method of Feasible Directions) to study the influence from numerical noise. Both optimisation methods were able to find the global optimum with the two different grid sizes (the search path for the gradient‐based method on the coarse grid was able to avoid the region in the design space containing local minima). However, the RSM needed fewer iterations in reaching the optimum. From a grid convergence study at two points in the design space the level of noise appeared to be sufficiently low, when the relative step size is 10–4 for the finite difference calculations, to not influence the convergence if the errors are below 5 per cent for this contraction geometry.

Design optimisation of a thermo-elastic-plastic process for plate forming

A design optimisation problem for the thermo-elastic-plastic process or the line heating (LH) process for plate forming is formulated. The deviations of position vectors from the objective shape are used as constraints. The objective function is to minimise the weight sum of these constraints. Design variables are torch velocity, torch strength, cooler distance, and plate thickness. An automated design optimisation procedure is developed using the method of feasible direction and the finite element solvers for heat convection and thermal stress analyses. The necessary gradient information is obtained using the forward finite difference method. During the LH process, the temperature field is assumed to be uncoupled with the displacement field. A numerical experiment shows that inelastic behaviour of the material is crucial to the residual stress during the LH process. Application of the design optimisation to a flat plate model shows applicability and effectiveness for the LH process.

Global Optimization of Composite Structures Using Triangular Patch Algorithm

Several design problems of composite structures are studied via a global optimizer based on attraction regions. MSC/NASTRAN is adopted for static and eigenvalue analysis. The method of modified feasible direction in DOT is used for local optimization. Through the review of global optimization algorithms, the triangular patch algorithm is selected because the algorithm is known to be efficient, robust and powerful for general nonlinear optimization problems. For general applicability, various mechanical properties are considered as design objectives; strain energy, eigenvalue, weight, displacement, and buckling load. In all cases considered, the triangular patch algorithm results in a lot of optimum points and useful design patterns, that are not easy by local algorithms or conventional global algorithms can be determined.

On Applied Nonlinear and Bilevel Programming or Pursuit-Evasion Games

Motivated by the benefits of discretization in optimal control problems, we consider the possibility of discretizing pursuit-evasion games. Two approaches are introduced. In the first approach, the solution of the necessary conditions of the continuous-time game is decomposed into ordinary optimal control problems that can be solved using discretization and nonlinear programming techniques. In the second approach, the game is discretized and transformed into a bilevel programming problem, which is solved using a first-order feasible direction method. Although the starting points of the approaches are different, they lead in practice to the same solution algorithm. We demonstrate the usability of the discretization by solving some open-loop representations of feedback solutions for a complex pursuit-evasion game between a realistically modeled aircraft and a missile, with terminal time as the payoff. The solutions are compared with those obtained via an indirect method.

A superlinearly convergent method of feasible directions

Method of feasible directions (MFD) is one of the most important methods for solving engineering optimization problems. There exist two types of MFD: one has at most a linear convergence rate and the other has a superlinear convergence rate. The former type includes Zoutendijk's MFD, Topkis–Veinott's MFD, Pironneau–Polak's MFD, Cawood–Kostreva's Norm-Relaxed MFD, and a modified MFD. The latter includes Panier–Tits' FSQP approaches. However, the price of the superlinearly convergent MFD for achieving superlinearity is rather high computational cost per single iteration. This paper presents an algorithm based on the modified MFD. The theoretical analysis shows that this algorithm is locally superlinearly convergent. This algorithm has attained the same convergence rate as FSQP while it only needs to solve two subproblems instead of three subproblems in FSQP. Hence, this algorithm should be computationally more efficient than FSQP.

Steepest descent methods for multicriteria optimization

We propose a steepest descent method for unconstrained multicriteria optimization and a “feasible descent direction” method for the constrained case. In the unconstrained case, the objective functions are assumed to be continuously differentiable. In the constrained case, objective and constraint functions are assumed to be Lipshitz-continuously differentiable and a constraint qualification is assumed. Under these conditions, it is shown that these methods converge to a point satisfying certain first-order necessary conditions for Pareto optimality. Both methods do not scalarize the original vector optimization problem. Neither ordering information nor weighting factors for the different objective functions are assumed to be known. In the single objective case, we retrieve the Steepest descent method and Zoutendijk's method of feasible directions, respectively.

Design sensitivity analysis and optimization of non-linear transient dynamics. Part II?configuration design

For configuration design of non-linear dynamic structures, such as crashworthiness design, a continuum-based configuration design sensitivity analysis (DSA) and optimization methods are developed. The same transient dynamic analysis method used in Part I of this paper is employed here. The elastic–plastic material and finite strain and rotation effect are considered. The first-order variations of the energy forms, load form and kinematic and structural responses with respect to configuration design variables are derived. For the configuration design, both the shape and orientation variations contribute to the first-order variation of the equations of motion. The angular design velocity field associated with Euler angles is used to represent the orientation variation to obtain accurate design sensitivity results. Like Part I of the paper, the updated Lagrangian formulation and direct differentiation method are used for DSA for the path-dependent problem. For the updated Lagrangian formulation, the design velocity field that defines the mapping between the initial and perturbed designs should also be updated at each configuration. Numerical implementation of configuration DSA and optimization is carried out for fixed time steps using DYNA3D and the modified feasible direction (MFD) method. It is observed that the proposed DSA method yields better sensitivity results than the finite difference method for the highly non-linear problem. Design optimization is carried out using the design sensitivity information.Copyright © 2000 John Wiley & Sons, Ltd.

On the numerical treatment of linearly constrained semi-infinite optimization problems

We consider the application of two primal algorithms to solve linear semi-infinite programming problems depending on a real parameter. Combining a simplex-type strategy with a feasible-direction scheme we obtain a descent algorithm which enables us to manage the degeneracy of the extreme points efficiently. The second algorithm runs a feasible-direction method first and then switches to the purification procedure. The linear programming subproblems that yield the search direction involve only a small subset of the constraints. These subsets are updated at each iteration using a multi-local optimization algorithm. Numerical test examples, taken from the literature in order to compare the numerical effort with other methods, show the efficiency of the proposed algorithms.

Configuration and sizing design optimisation of powertrain mounting systems

A configuration and sizing design optimisation problem for powertrain mounting systems is formulated. The objective function is to decouple and maximise the component modal kinetic energy of the powertrain subsystem. Natural frequencies and collinear roll vector condition are used as the constraints. Cartesian coordinates and spring rates of the mounts are selected as the design parameters. An automated design optimisation procedure is developed using an optimisation code based on the method of feasible direction (MFD) and MSC/NASTRAN as a finite clement solver. The necessary gradient information is obtained using the central finite difference method. For the obtained optimum model, forced response analyses are performed in engine idling and wheel unbalancing conditions to show that overall accelerations are greatly reduced at the response positions.

Robust control via concave minimization local and global algorithms

This paper is concerned with the robust control problem of linear fractional representation (LFT) uncertain systems depending on a time-varying parameter uncertainty. Our main result exploits a linear matrix inequality (LMI) characterization involving scalings and Lyapunov variables subject to an additional essentially nonconvex algebraic constraint. The nonconvexity enters the problem in the form of a rank deficiency condition or matrix inverse relation on the scalings only. It is shown that such problems, but also more generally rank inequalities and bilinear constraints, can be formulated as the minimization of a concave functional subject to LMI constraints. First of all, a local Frank and Wolfe (1956) feasible direction algorithm is introduced in this context to tackle this hard optimization problem. Exploiting the attractive concavity structure of the problem, several efficient global concave programming methods are then introduced and combined with the local feasible direction method to secure and certify global optimality of the solutions. Computational experiments indicate the viability of our algorithms, and in the worst case, they require the solution of a few LMI programs.

Optimal design of steel frames subject to gravity and seismic codes' prescribed lateral forces

Allowable stress design of two-dimensional braced and unbraced steel frames based on AISC specifications subject to gravity and seismic lateral forces is formulated as a structural optimization problem. The nonlinear constrained minimization algorithm employed is the feasible directions method. The objective function is the weight of the structure, and behaviour constraints include combined bending and axial stress, shear stress, buckling, slenderness, and drift. Cross-sectional areas are used as design variables. The anylsis is performed using stiffness formulation of the finite element analysis method. Equivalent static force and response spectrum analysis methods of seismic codes are considered. Based on the suggested methodology, the computer program OPTEQ has been developed. Examples are presented to illustrate the capability of the optimal design approach in comparative study of various types of frames subjected to gravity loads and seismic forces according to a typical code.

A generalization of the norm-relaxed method of feasible directions

This paper generalizes the norm-relaxed method of feasible directions which has roots in the classical method of feasible directions (MFD). This generalized algorithm introduces some parameters which can be adjusted to speed up the convergence. An investigation of its global convergence is included and the numerical performance is shown on 15 test problems. Its global convergence is guaranteed under a rather mild and standard assumption. The numerical tests show that this new algorithm converges faster than the previous norm-relaxed MFD for most test problems by selecting a set of suitable parameters even though it only has the same guaranteed convergence rate as the previous norm-relaxed MFD.

An Optimization Method for Design of Subambient Pressure Shaped Rail Sliders

This study proposes a design methodology for determining configurations of subamient pressure shaped rail sliders by using a nonlinear programming technique in order to meet the desired flying characteristics over the entire recording band. The desired flying characteristics considered in this study are to minimize the variation in flying height from a target value, to keep the pitch angle within a suitable range, and to ensure that the outside rail flies lower than the inside rail even with the roll distribution due to manufacturing process. The design variables selected are recess depth, geometry of the air bearing surface, and pivot location in the transverse direction of the slider. The method of feasible directions in Automated Design Synthesis (ADS) is utilized to automatically find the optimum design variables which simultaneously meet all the desired flying characteristics. To validate the suggested design methodology, a computer program is developed and applied to a 30 percent/15 nm twin rail slider and a 30 percent/15 nm tri-rail slider. Simulation results for both sliders demonstrated the effectiveness of the proposed design methodology by showing that the flying characteristics of the optimally designed sliders are enhanced in comparison with those of the initial ones.

Global Convergence Analysis of Algorithms for Finding Feasible Points in Norm-Relaxed MFD

This paper introduces two new algorithms for finding initial feasible points from initial infeasible points for the recently developed norm-relaxed method of feasible directions (MFD). Their global convergence is analyzed. The theoretical results show that both methods are globally convergent; one of them guarantees finding a feasible point in a finite number of steps. These two methods are very convenient to implement in the norm-relaxed MFD. Numerical experiments are carried out to demonstrate their performance on some classical test problems and to compare them with the traditional method of phase I problems. The numerical results show that the methods proposed in this paper are more effective than the method of phase I problems in the norm-relaxed MFD. Hence, they can be used for finding initial feasible points for other MFD algorithms and other nonlinear programming methods.

An optimum design of the sub-ambient pressure shaped rail sliders on flying characteristics considering the high altitude condition

We present optimum results of the two sub-ambient pressure shaped rail sliders. The goal of optimization is to meet the desired flying characteristics over the entire data zone at sea level as well as at the 5000 ft. Altitude condition. The design variables are the geometry of the air bearing slider (ABS), the recess depth, and the location of the pivot. The method of feasible direction is utilized to find the optimum designs. The optimally designed sliders show the enhanced flying characteristics along with the less deviation at the high altitude condition.

Aerodynamic Shape Optimization of a Subsonic Inlet Using Three-Dimensional Euler Computation

A design optimization study of a subsonic inlet is presented where the peak Mach number at cruise condition on the inside of the inlet surface is the objective function to be minimized and inlet lip shape parameters are the design variables. The peak Mach number at takeoff, rolling takeoff, static, and crosswind conditions are constrained to an upper limit. The three-dimensional  owŽ eld is predicted by the NPARC computational  uid dynamics code in Euler mode. A GRAPE(grids about airfoils using Poisson’s equation) based three-dimensional grid generator is employed to generate a C-grid around the inlet. Constrained numerical optimization is carried out by interfacing NPARC with the automated design synthesis optimization code. The constrained optimum is obtained by the method of feasible directions. The required gradients for optimization are computed via forward difference scheme. At the baseline (initial) design, takeoff, rolling takeoff, and crosswind conditions have their peak Mach numbers exceeding the upper bounds. However, at the constrained optimum design, only the upper limit on the takeoff peak Mach number is found to be critical with no appreciable change in cruise Mach number.

The Minimum Covering l_pb -Hypersphere Problem

The minimu vering hypersphere problem is defined as to find a hypersphere of minimum radius enclosing a finite set of given points in R^n. A hypersphere is a setS(c,r)={x ∈ R^n : d(x,c) ≤ r} , where c is the center of S, r is the radius of S and d(x,c) is the Euclidean distance between x and c, i.e.,d(x,c)=l_2(x-c) . We consider the extension of this problem when d(x, c) is given by anyl_pb -norm, where 1 0,j=1,...,n , then S(c,r) is called anl_pb -hypersphere, in particular for p=2 and b_j=1, j=1,...,n, we obtain thel_2 -norm. We study some properties and propose some primal and dual algorithms for the extended problem , which are based on the feasible directions method and on the Wolfe duality theory. By computational experiments, we compare the proposed algorithms and show that they can be used to approximate the smallestl_pb -hypersphere enclosing a large set of points inR^n .

A study on the machining-parameters optimization of wire electrical discharge machining

A suitable selection of machining parameters for the wire electrical discharge machining (WEDM) process relies heavily on the operators' technologies and experience because of their numerous and diverse range. Machining-parameters tables provided by the machine-tool builder can not meet the operators' requirements, since for an arbitrary desired roughness of the machining surface, they do not provide the optimal machining conditions. An approach to determine parameters setting is proposed. Based on the Taguchi quality design method and the analysis of variance, the significant factors affecting the machining performance such as metal removal rate, gap width, surface roughness, sparking frequency, average gap voltage and normal ratio (ratio of normal sparks to total sparks) are determined. By means of regression analysis, mathematical models relating the machining performance and various machining parameters are established. Based on the mathematical models developed, an objective function under the multi-constraint conditions is formulated. The optimization problem is solved by the feasible direction method, and the optimal machining parameters are obtained. Experimental results demonstrate that the machining models are appropriate and the derived machining parameters satisfy the real requirements in practice.