Direct Optimization Approach Research Articles

Designing resilient networks using multicriteria metaheuristics

The paper deals with the design of resilient networks that are fault tolerant against link failures. Usually, fault tolerance is achieved by providing backup paths, which are used in case of an edge failure on a primary path. We consider this task as a multiobjective optimization problem: to provide resilience in networks while minimizing the cost subject to capacity constraint. We propose a stochastic approach, which can generate multiple Pareto solutions in a single run. The feasibility of the proposed method is illustrated by considering several network design problems using a single weighted average of objectives and a direct multiobjective optimization approach using the Pareto dominance concept.

A nonlinear nonconvex minimum total heat transfer area formulation for ocean thermal energy conversion (OTEC) systems

In this paper a technical analysis of an ocean thermal energy conversion (OTEC) system is performed. Specifically, we present a general mathematical framework for the synthesis of OTEC power generating systems. The overall synthesis task is to minimize heat exchange area requirements, while generating some fraction of the maximum net power recoverable from hot and cold ocean water. The resulting problem formulation yields a nonlinear, nonconvex mathematical program; however, we show that globally optimal solutions for this program are easily obtained explicitly through a direct optimization approach with minimal computational effort over a wide range of thermodynamic conditions. The proposed analysis is demonstrated on a case study involving the generation of hydrogen by an OTEC system with a pure ammonia working fluid.

Direct optimization approach for lithographic process conditions

We present a semiautomatic procedure for the optimization of lithographic process conditions. In the regime of resolution enhancement techniques such as optical proximity correction, off-axis illumination, and phase shifting masks, the design of lithographic photomasks and illumination sources becomes increasingly intricate. Earlier "what you see is what you get" (WYSIWYG) approaches cannot be applied anymore. Instead, the employment of sophisticated design optimization tools is inevitable. In contrast to related efforts using an inverse problem formulation to address this goal, the approach presented here can be considered a direct method: mask and illumination layouts are mutually altered, evaluated, and improved, using an heuristic search algorithm. Thus, not only does this method require very little a priori knowledge of the process, it also allows for a very flexible problem formulation, enabling an easy integration of different model options or even process steps. As an underlying optimization algorithm of the presented procedure, a genetic algorithm has been used, whose flow, data representation, and basic operations are discussed briefly. Different representation types for both mask and illumination setups and the formulation of the optimization objectives are explained in detail. Various well-performing mask and illumination settings demonstrate the feasibility of the proposed approach and point out further potentials: in contrast to single problem specific approaches, this method is applicable to a great variety of different complex optimization tasks in advanced lithography.

Alignment and Topological Accuracy of the Direct Optimization approach via POY and Traditional Phylogenetics via ClustalW + PAUP*

Direct optimization frameworks for simultaneously estimating alignments and phylogenies have recently been developed. One such method, implemented in the program POY, is becoming more common for analyses of variable length sequences (e.g., analyses using ribosomal genes) and for combined evidence analyses (morphology + multiple genes). Simulation of sequences containing insertion and deletion events was performed in order to directly compare a widely used method of multiple sequence alignment (ClustalW) and subsequent parsimony analysis in PAUP* with direct optimization via POY. Data sets were simulated for pectinate, balanced, and random tree shapes under different conditions (clocklike, non-clocklike, and ultrametric). Alignment accuracy scores for the implied alignments from POY and the multiple sequence alignments from ClustalW were calculated and compared. In almost all cases (99.95%), ClustalW produced more accurate alignments than POY-implied alignments, judged by the proportion of correctly identified homologous sites. Topological accuracy (distance to the true tree) for POY topologies and topologies generated under parsimony in PAUP* from the ClustalW alignments were also compared. In 44.94% of the cases, Clustal alignment tree reconstructions via PAUP* were more accurate than POY, whereas in 16.71% of the cases POY reconstructions were more topologically accurate (38.38% of the time they were equally accurate). Comparisons between POY hypothesized alignments and the true alignments indicated that, on average, as alignment error increased, topological accuracy decreased.

Optimization-based trajectory planning of the human upper body

This paper presents studies of the coordination of human upper body voluntary movement. A minimum-jerk 3D model is used to obtain the desired path in Cartesian space, which is widely used in the prediction of human reach movement. Instead of inverse kinematics, a direct optimization approach is used to predict each joint's profile (a spline curve). This optimization problem has four cost function terms: (1) Joint displacement function that evaluates displacement of each joint away from its neutral position; (2) Inconsistency function, which is the joint rate change (first derivative) and predicted overall trend from the initial target point to the final target point; (3) The non-smoothness function of the trajectory, which is the second derivative of the joint trajectory; (4) The non-continuity function, which consists of the amplitudes of joint angle rates at the initial and final target points, in order to emphasize smooth starting and ending conditions. This direct optimization technique can be used for potentially any number of degrees of freedom (DOF) system and it reduces the cost associated with certain inverse kinematics approaches for resolving joint profiles. This paper presents a high redundant upper-body modeling with 15 DOFs. Illustrative examples are presented and an interface is set up to visualize the results.

Optimal strictly positive real controllers using direct optimization

In this paper, we consider the H 2 -optimal control problem subject to the constraint that the resulting controller be strictly positive real. A direct numerical optimization approach is adopted in conjunction with a controller parametrization that is linear in the unknown parameters. The SPR constraint is easily expressed at each frequency in the form of a linear inequality. The method is applied to a numerical example from the literature and good results are achieved. In particular, the proposed method is particularly adept at determining low order controllers.

DIRECT WEIGHT OPTIMIZATION FOR APPROXIMATELY LINEAR FUNCTIONS: OPTIMALITY AND DESIGN

The Direct Weight Optimization (DWO) approach to estimating a regression function is studied here for the class of approximately linear functions, i.e., functions whose deviation from an affine function is bounded by a known constant. Upper and lower bounds for the asymptotic maximum MSE are given, some of which also hold in the non-asymptotic case and for an arbitrary fixed design. Their coincidence is then studied. Particularly, under mild conditions, it can be shown that there is always an interval in which the DWO-optimal estimator is optimal among all estimators. Experiment design issues are also studied.

Analytic energy gradients of the optimized effective potential method

The analytic energy gradients of the optimized effective potential (OEP) method in density-functional theory are developed. Their implementation in the direct optimization approach of Yang and Wu [Phys. Rev. Lett. 89, 143002 (2002)] and Wu and Yang [J. Theor. Comput. Chem. 2, 627 (2003)] are carried out and the validity is confirmed by comparison with corresponding gradients calculated via numerical finite difference. These gradients are then used to perform geometry optimizations on a test set of molecules. It is found that exchange-only OEP (EXX) molecular geometries are very close to the Hartree-Fock results and that the difference between the B3LYP and OEP-B3LYP results is negligible. When the energy is expressed in terms of a functional of Kohn-Sham orbitals, or in terms of a Kohn-Sham potential, the OEP becomes the only way to perform density-functional calculations and the present development in the OEP method should play an important role in the applications of orbital or potential functionals.

A GENERAL DIRECT WEIGHT OPTIMIZATION FRAMEWORK FOR NONLINEAR SYSTEM IDENTIFICATION

The direct weight optimization (DWO) approach is a method for finding optimal function estimates via convex optimization, applicable to nonlinear system identification. In this paper, an extended version of the DWO approach is introduced. A general function class description — which includes several important special cases — is presented, and different examples are given. A general theorem about the principal shape of the weights is also proven.

Multi-Objective Optimisation of Restricted Complexity Controllers*

Restricted complexity controller design for the active suspension benchmark problem call for papers EJC special issue on Design and Optimisation of Restricted Complexity Controllers. Web site http://www-ejc.ensieg.inpg.fr/ , 2002 is considered. The control design specifications of the benchmark is firstly recast into a mixed H ∞ /l 1 optimisation problem, which is solved via a convex optimisation approach. A globally optimal Pareto curve is produced via the optimisation. It presents the limits of performance and is served as a reference for restricted complexity controller design. Then, controllers with different complexity are designed via a direct optimisation approach. The performance indices of these controllers are compared with the global Pareto curve. Based on the comparison, the controller with three parameters is determined as the one achieving acceptable performance with lowest complexity. Experimental results on the real system confirm the satisfactory performance achieved by the controller.

Constraint resolution using optimisation techniques

Constraint-based CAD systems are useful for handling design applications in which the precise rules are not known beforehand. The designer can introduce new constraints as these become apparent as the design task proceeds. A variety of ways for resolving constraints currently exist. This paper considers the use of a direct search optimisation approach. It is shown that this has the advantage of being able to handle a variety of different applications and that problems of under- or over-constrained systems no longer exist. A software system using this approach is introduced and its application to a number of areas is illustrated, including typical problems from computer-aided sketching, assembling modelling and mechanism simulation.

A Direct Optimization Approach to Hidden Markov Modeling for Single Channel Kinetics

Hidden Markov modeling (HMM) provides an effective approach for modeling single channel kinetics. Standard HMM is based on Baum's reestimation. As applied to single channel currents, the algorithm has the inability to optimize the rate constants directly. We present here an alternative approach by considering the problem as a general optimization problem. The quasi-Newton method is used for searching the likelihood surface. The analytical derivatives of the likelihood function are derived, thereby maximizing the efficiency of the optimization. Because the rate constants are optimized directly, the approach has advantages such as the allowance for model constraints and the ability to simultaneously fit multiple data sets obtained at different experimental conditions. Numerical examples are presented to illustrate the performance of the algorithm. Comparisons with Baum's reestimation suggest that the approach has a superior convergence speed when the likelihood surface is poorly defined due to, for example, a low signal-to-noise ratio or the aggregation of multiple states having identical conductances.

Optimal sparing for SF 6 gas insulated substations

A method is suggested to optimize the number of spare items of substation components which have to be stored at the beginning of each year throughout a planning period. This sparing policy is conceived to provide the minimum total cost consisting of investment and load curtailment costs. The method uses the Hooke and Jeeves' direct search optimization approach. The optimization method is extended to apply to typical HV/MV GIS transformer stations requiring specific reliability modeling.

Optimal multirate control design for hard disk drive servo systems

This paper presents a multirate digital controller design for HDD servo systems via direct parameter optimization approach. In order to find a satisfactory multirate controller with good transient and steady state performance, a linear quadratic performance index with stability constraints is introduced. The controller parameters are selected based on the HDD servo system transient response data instead of a detailed plant model. The proposed method can improve the performance of LQG multirate servo designs without knowing the exact plant parameters or noise model. The effect of controller order on system performance is also studied. For a 4th order system studied, it is found that a 4th order controller is sufficient to achieve the optimal performance.

Stepping over obstacles during locomotion: insights from multiobjective optimization on set of input parameters.

In this study we investigate possible objectives that the central nervous system (CNS) may consider in planning a strategy for stepping over an obstacle. A link segment simulation model has been developed based on Lagrangian dynamics, with which muscle force inputs can be optimized to best satisfy the postulated objectives for landing stability, obstacle clearance, and efficiency of the movement. A direct optimization approach with multiobjective criteria based on the kinematic and kinetic characteristics of the swing phase of locomotion is used in the simulation. The role of initial conditions at toe-off and biarticular muscle forces during the swing phase was also investigated. The optimization was performed for both leading limb and the trailing limb during the swing phase. The simulation results demonstrate that the use of biarticular muscles is sufficient to clear a range of obstacles with the trailing limb (obstacle encountered during early swing). Stride length or landing stability objectives need not be specified suggesting a simpler control of trailing limb trajectory by the CNS (one of stride length or landing stability objectives were not necessary). In contrast while the use of biarticular muscles can be sufficient to clear obstacles with the leading limb (obstacle encountered during mid to late swing), a stable landing and smooth toe and knee trajectories are compromised without suitable initial conditions at toe-off. The results suggest that the set of postulated objectives for the lead limb is adequate, although not complete.

Method for automatic costate calculation

A method for the automatic calculation of costates using only the results obtained from direct optimization techniques is presented. The approach exploits the relation between the time-varying costates and certain sensitivities of the variational cost function, a relation that also exists between the Lagrangian multipliers obtained from a direct optimization approach and the sensitivities of the associated nonlinear-programming cost function. The complete theory for treating free, control-constrained, interior-point-constrained, and state-constrained optimal control problems is presented. As a numerical example, a state-constrained version of the brachistochrone problem is solved and the results are compared to the optimal solution obtained from Pontryagin's minimum principle. The agreement is found to be excellent. Nomenclature / = right-hand side of state equations ge = control equality constraints gi = control inequality constraints he = state equality constraints hi = state inequality constraints J = cost function M = interior-point constraints m = dimension of control vector u N = total number of nodes minus 1 = total number of subintervals n — dimension of state vector x PWC = set of piecewise continuous functions t = time tf = final time ti = nodes along the time axis to = initial time u = control vector x = state vector Xf = final state Xi = state vector at node £/ Xo = initial state \(t) = costate A/ = Lagrangian multiplier associated with differential constraints along subinterval / Hi - Lagrangian multiplier associated with state constraints at node i (Ti = Lagrangian multiplier associated with control constraints along subinterval / <£ = cost function if} j. = boundary conditions at final time •00 = boundary conditions at initial time

A direct optimization approach to the two-dimensional acoustic inverse problem in the time domain

The acoustic inverse problem on a stratified half-space is considered in the time domain. By constructing an objective functional, the inverse problem is thereby reduced to an optimization problem where the profiles of the velocity and the density minimizing the functional are to be found. A direct optimization algorithm is employed to solve the optimization problem and reconstructed the profiles of the velocity and the density simultaneously. The stability theorem and the numerical examples are given. The reconstruction results excellently agree with the true profiles.

Dextrous hand grasping force optimization

A key goal in dextrous robotic hand grasping is to balance external forces and at the same time achieve grasp stability and minimum grasping energy by choosing an appropriate set of internal grasping forces. Since it appears that there is no direct algebraic optimization approach, a recursive optimization, which is adaptive for application in a dynamic environment, is required. One key observation in this paper is that friction force limit constraints and force balancing constraints are equivalent to the positive definiteness of a certain matrix subject to linear constraints. Based on this observation, we formulate the task of grasping force optimization as an optimization problem on the smooth manifold of linearly constrained positive definite matrices for which there are known globally exponentially convergent solutions via gradient flows. There are a number of versions depending on the Riemannian metric chosen, each with its advantages, Schemes involving second derivative information for quadratic convergence are also studied. Several forms of constrained gradient flows are developed for point contact and soft-finger contact friction models. The physical meaning of the cost index used for the gradient flows is discussed in the context of grasping force optimization. A discretized version for real-time applicability is presented. Numerical examples demonstrate the simplicity, the good numerical properties, and optimality of the approach.

Variational solutions for the heat-rate-limited aeroassisted orbital transfer problem

Minimum energy loss trajectories based on Pontryagin's minimum principle are generated for the atmospheric part of an aeroassisted orbital transfer problem. At the initial time all states are given. At the final time the altitude, velocity, and inclination change are prescribed. Additionally, the trajectory is subject to a heating rate limit, which, in the context of optimal control, represents a first-order state inequality constraint. Numerical difficulties implied by the singularly perturbed nature of the problem are avoided by starting the integration in the interior of the trajectory at a point where the heating rate limit is active. The state constraint becomes active always in the form of a state constrained arc. Nontrivial touch points have not been observed in numerical solutions. VER the years a considerable amount of research has been conducted in the area of aeroassisted orbital transfers (AOT). Surveys on the current state of the art in optimization of AOT trajectories can be found in Refs. 1 and 2. The optimization of the atmospheric part of orbital plane changes using aerodynamic controls is discussed in Refs. 3-6. In Ref. 5, a guidance law is developed through regular expansion of the Hamilton-Jacobi-Bellmann equation. A guidance law development through matched asymptotic expansions and through a hybrid regular expansion/collocation approach is presented in Refs. 7 and 8, respectively. In Ref. 9, maximum cross range and maximum orbital plane change trajectories are calculated using a nonlinear programming code. In Ref. 10, the use of multiple pass trajectories is investigated to alleviate the severe aeroheating. In Ref. 11, aerocruise and aeroglide maneuvers are compared, both in presence of a heating rate limit. Recently, variational solutions for the heating rate limited aerocruise maneuver were presented in Ref. 12. Numerical difficulties were reported, and converged solutions could not be obtained below a certain value for the prescribed maximum heating rate. The optimal switching structure was identified as (free, touch point, free). Solutions with state constrained arcs were not obtained. For the same vehicle model and atmospheric model as in Ref. 12 the heating rate limit could be reduced much further in Ref. 13, using direct shooting and direct collocation. The obtained trajectories are physically feasible and involve state constrained arcs. The solutions were reported to be very sensitive, and certain convergence difficulties were overcome by reducing the prescribed precision in the nonlinear program solver. The present paper is based on a dynamical model that is identical to the one used in Ref. 12. The objective is to verify the results reported in Ref. 12, to identify the correct optimal switching structure, and to generate variational benchmark solutions that could be used to verify and to calibrate direct optimization approaches.

A unified approach to statistical tomography using coordinate descent optimization

Over the past years there has been considerable interest in statistically optimal reconstruction of cross-sectional images from tomographic data. In particular, a variety of such algorithms have been proposed for maximum a posteriori (MAP) reconstruction from emission tomographic data. While MAP estimation requires the solution of an optimization problem, most existing reconstruction algorithms take an indirect approach based on the expectation maximization (EM) algorithm. We propose a new approach to statistically optimal image reconstruction based on direct optimization of the MAP criterion. The key to this direct optimization approach is greedy pixel-wise computations known as iterative coordinate decent (ICD). We propose a novel method for computing the ICD updates, which we call ICD/Newton-Raphson. We show that ICD/Newton-Raphson requires approximately the same amount of computation per iteration as EM-based approaches, but the new method converges much more rapidly (in our experiments, typically five to ten iterations). Other advantages of the ICD/Newton-Raphson method are that it is easily applied to MAP estimation of transmission tomograms, and typical convex constraints, such as positivity, are easily incorporated.