Active Set Strategy Research Articles

Stress constrained topology optimization with free-form design domains

This paper aims at dealing with realistic and challenging design problems of stress constrained topology optimization with free-form design domains. First, the concept of level set function (LSF) based modelers is introduced to transform this kind of problems into the Boolean conjunction operation of a topology variation modeler (TVM) onto a free-form design domain modeler (FDDM). Such an operation is mathematically realized by means of the so-called R-functions in the form of implicit LSFs. Within this framework, topology optimization problems are classified into two general cases depending upon the existence of non-designable solid feature. Analytical sensitivity analysis formulas are further derived. Compared with the existing level set based method, the important sensitivity property of design domain preserving makes it possible to avoid automatically the boundary violation of the design domain caused by the zero level set movement and both the topology and boundary shape of the free-form design domain can be simultaneously optimized. Second, the implementation of the finite cell method (FCM) ensures the stress computing accuracy in the fixed mesh due to the use of high-order shape functions and adaptive integration scheme. The combination of the active-set strategy and the dynamic aggregation technique also reduces the number of local stress constraints greatly. Finally, representative examples are presented to illustrate the conveniences and effectiveness of the proposed method.

Active-set strategy in Powell's method for optimization without derivatives

In this article we present an algorithm for solving bound constrained optimization problems without derivatives based on Powell's method [38] for derivative-free optimization. First we consider the unconstrained optimization problem. At each iteration a quadratic interpolation model of the objective function is constructed around the current iterate and this model is minimized to obtain a new trial point. The whole process is embedded within a trust-region framework. Our algorithm uses infinity norm instead of the Euclidean norm and we solve a box constrained quadratic subproblem using an active-set strategy to explore faces of the box. Therefore, a bound constrained optimization algorithm is easily extended. We compare our im_ plementation with NEWUOA and BOBYQA, Powell's algorithms for unconstrained and bound constrained derivative free optimization respectively. Numerical experiments show that, in general, our algorithm require less functional evaluations than Powell's algorithms.

Active set strategy of optimized extreme learning machine

Extreme learning machine (ELM) has been introduced as a simple and efficient learning approach for regression and classification applications. From the optimization point of view, optimized ELM is equivalent to SVM, but with less constraints in the optimization formulation and random ELM kernel. This paper introduces an active set based optimized ELM approach to solve bound constrained optimization problem in a straightforward way, which operates on a small working set of variables at each iteration. Thus, the constrained problem can be eventually solved by an unconstrained algorithm, and this enables us to establish a global convergence theory. The approach requires less time for quadratic programming solving and provides better generalization performance. In addition, the proposed approach with much smaller number of non-bound support values is significantly faster than SVM with active set strategy for large training data set.

Augmented Lagrangian Method for Nonlinear Programming with many Complex Constraints

A method of multiplier is presented for solving optimization problems. For large-scale constraint problems, combining the active set strategy, we use the aggregate function to approximate the max-value function. Only a few of functions are involved at each iteration, so the computation for gradient is significantly reduced. The numerical results show that the method is effective.

A general optimality criteria algorithm for a class of engineering optimization problems

An optimality criteria (OC)-based algorithm for optimization of a general class of nonlinear programming (NLP) problems is presented. The algorithm is only applicable to problems where the objective and constraint functions satisfy certain monotonicity properties. For multiply constrained problems which satisfy these assumptions, the algorithm is attractive compared with existing NLP methods as well as prevalent OC methods, as the latter involve computationally expensive active set and step-size control strategies. The fixed point algorithm presented here is applicable not only to structural optimization problems but also to certain problems as occur in resource allocation and inventory models. Convergence aspects are discussed. The fixed point update or resizing formula is given physical significance, which brings out a strength and trim feature. The number of function evaluations remains independent of the number of variables, allowing the efficient solution of problems with large number of variables.

A new active-set strategy for NCP with degenerate solutions

In this paper, we present a semismooth Newton-type method for solving nonlinear complementarity problems with degenerate solutions. At each iteration, an estimated set is used to approximate the degenerate indices of the solutions. Then, a judge step is introduced to decide whether a standard Newton step or a reduced Newton step will be executed. The method combines the standard step and the reduced step in one algorithm. The standard Newton step is mainly used to keep the rate of the global convergence. When near the solution, the degenerate indices will be correctly identified and the original problem will be transformed to a reduced non-degenerate problem. Under some mild assumptions, local superlinear convergence is obtained.

A variable fixing version of the two-block nonlinear constrained Gauss–Seidel algorithm for $$\ell _1$$ ℓ 1 -regularized least-squares

The problem of finding sparse solutions to underdetermined systems of linear equations is very common in many fields as e.g. signal/image processing and statistics. A standard tool for dealing with sparse recovery is the $$\ell _1$$ l 1 -regularized least-squares approach that has recently attracted the attention of many researchers. In this paper, we describe a new version of the two-block nonlinear constrained Gauss---Seidel algorithm for solving $$\ell _1$$ l 1 -regularized least-squares that at each step of the iteration process fixes some variables to zero according to a simple active-set strategy. We prove the global convergence of the new algorithm and we show its efficiency reporting the results of some preliminary numerical experiments.

A frictional mortar contact approach for the analysis of large inelastic deformation problems

A multibody frictional mortar contact formulation (Gitterle et al., 2010) is extended for the simulation of solids undergoing finite strains with inelastic material behavior. The framework includes the modeling of finite strain inelastic deformation, the numerical treatment of frictional contact conditions and specific finite element technology. Several well-established and recent models are employed for each of these building blocks to capture the distinct physical aspects of the deformation behavior. The approach is based on a mortar formulation and the enforcement of contact constraints is realized with dual Lagrange multipliers. The introduction of nonlinear complementarity functions into the frictional contact conditions combined with the global equilibrium leads to a system of nonlinear equations, which is solved in terms of the semi-smooth Newton method. The resulting method can be interpreted as a primal–dual active set strategy (PDASS) which deals with contact nonlinearities, material and geometrical nonlinearities in one iterative scheme. The consistent linearization of all building blocks of the framework yields a robust and highly efficient approach for the analysis of metal forming problems. The effect of finite inelastic strains on the solution behavior of the PDASS method is examined in detail based on the complementarity parameters. A comprehensive set of numerical examples is presented to demonstrate the accuracy and efficiency of the approach against the traditional node-to-segment penalty contact formulation.

A Primal Dual Active Set Algorithm With Continuation for Compressed Sensing

The success of compressed sensing relies essentially on the ability to efficiently find an approximately sparse solution to an under-determined linear system. In this paper, we developed an efficient algorithm for the sparsity promoting l <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> -regularized least squares problem by coupling the primal dual active set strategy with a continuation technique (on the regularization parameter). In the active set strategy, we first determine the active set from primal and dual variables, and then update the primal and dual variables by solving a low-dimensional least square problem on the active set, which makes the algorithm very efficient. The continuation technique globalizes the convergence of the algorithm, with provable global convergence under restricted isometry property (RIP). Further, we adopt two alternative methods, i.e., a modified discrepancy principle and a Bayesian information criterion, to choose the regularization parameter automatically. Numerical experiments indicate that our algorithm is very competitive with state-of-the-art algorithms in terms of accuracy and efficiency, without a priori information about the regularization parameter.

Proximal methods for the latent group lasso penalty

We consider a regularized least squares problem, with regularization by structured sparsity-inducing norms, which extend the usual l 1 and the group lasso penalty, by allowing the subsets to overlap. Such regularizations lead to nonsmooth problems that are difficult to optimize, and we propose in this paper a suitable version of an accelerated proximal method to solve them. We prove convergence of a nested procedure, obtained composing an accelerated proximal method with an inner algorithm for computing the proximity operator. By exploiting the geometrical properties of the penalty, we devise a new active set strategy, thanks to which the inner iteration is relatively fast, thus guaranteeing good computational performances of the overall algorithm. Our approach allows to deal with high dimensional problems without pre-processing for dimensionality reduction, leading to better computational and prediction performances with respect to the state-of-the art methods, as shown empirically both on toy and real data.

A variational approach to sparsity optimization based on Lagrange multiplier theory

Sparsity optimization for linear least squares problems formulated as non-smooth regularization problems are considered in infinite-dimensional sequence spaces ℓp with p ∈ [0, 1]. Necessary optimality conditions in the format of a complementarity system are obtained. A monotonically convergent scheme is developed for the case p ∈ (0, 1]. For the case p = 0 a primal dual active set strategy based on the Lagrange multiplier rule is proposed and analyzed for special cases.

The Bound-Constrained Conjugate Gradient Method for Non-negative Matrices

Existing conjugate gradient (CG)-based methods for convex quadratic programs with bound constraints require many iterations for solving elastic contact problems. These algorithms are too cautious in expanding the active set and are hampered by frequent restarting of the CG iteration. We propose a new algorithm called the Bound-Constrained Conjugate Gradient method (BCCG). It combines the CG method with an active-set strategy, which truncates variables crossing their bounds and continues (using the Polak---Ribiere formula) instead of restarting CG. We provide a case with n=3 that demonstrates that this method may fail on general cases, but we conjecture that it always works if the system matrix A is non-negative. Numerical results demonstrate the effectiveness of the method for large-scale elastic contact problems.

A Barzilai–Borwein type method for stochastic linear complementarity problems

We consider the expected residual minimization (ERM) formulation of stochastic linear complementarity problem (SLCP). By employing the Barzilai---Borwein (BB) stepsize and active set strategy, we present a BB type method for solving the ERM problem. The global convergence of the proposed method is proved under mild conditions. Preliminary numerical results show that the method is promising.

PARAMETRIC SYSTEM IDENTIFICATION APPLIED TO THE BOW SHOCK WAVE SHAPE

In this paper it is addressed the prediction of form and location of detached shock waves ahead of two-dimensional and axially symmetric bodies at an zero angle of attack. As shown in Figs. 5 and 6, results show a very good agreement with experimental data. In this context an approximate method, based on a simplified form of the continuity relation, is developed to predict the location of detached shock waves ahead of two-dimensional and axially symmetric bodies. In order to reduce the problem to an equivalent onedimensional form, it is assumed that: (1) The form of the shock between its foremost point and its sonic point is adequately represented by a hyperbola asymptotic to tile free-stream Mach lines; and (2) the sonic line between the shock and the body is straight and inclined at a constant angle. Although the new methodology has some points of contact with earlier methodologies, the novelty here is that it is used Missile Datcom code as an aid to find out sonic point on body and also it is adopted Parametric System Identification (PSI) in the determination of bow shock shape which uses the Matlab® optimizer fmincon function and an active set strategy to minimize an error in a rms sense subject to simple constraint placed on the parameters by the user. The optimizer function calls a user written function which calculates the shape of the shock wave using the current parameters supplied by optimizer. Also for the shock distance L, the methodology presented here allows to select the value of the mentioned constant angle consistently based either in aerodynamics literature or through physical considerations. As the L value is previously known from measurements or aerodynamics literature, it was used an optimizer to minimize the error between predicted and known result varying a parameter which absorbs all inconsistencies that arise when it is used the basic Moeckel’s model considered here. Once the principal characteristics of the shock wave are calculated, an error value is returned to optimizer function based on the differences between predicted and known results.

Boundary element methods for variational inequalities

In this paper we present a priori error estimates for the Galerkin solution of variational inequalities which are formulated in fractional Sobolev trace spaces, i.e. in $$\widetilde{H}^{1/2}(\Gamma )$$ H ? 1 / 2 ( Γ ) . In addition to error estimates in the energy norm we also provide, by applying the Aubin---Nitsche trick for variational inequalities, error estimates in lower order Sobolev spaces including $$L_2(\Gamma )$$ L 2 ( Γ ) . The resulting discrete variational inequality is solved by using a semi-smooth Newton method, which is equivalent to an active set strategy. A numerical example is given which confirms the theoretical results.

Chebyshev fitting of complex surfaces for precision metrology

The form qualities of precision components are essential for their functionalities. The Peak-to-Valley parameters are widely adopted to assess the form accuracies of optical components. The commonly used least squares method is prone to over-estimation, thus the Chebyshev fitting should in turn be implemented. In this paper the original minimax optimisation problem is converted into an unconstrained differentiable minimisation problem by the exponential penalty functions. The fitting accuracy and numerical stability are balanced by employing an active-set strategy and adjusting the configuration parameters adaptively. Finally some benchmark data sets are applied to demonstrate the validity and efficiency of this method.

Stress-based topology optimization of concrete structures with prestressing reinforcements

Following the extended two-material density penalization scheme, a stress-based topology optimization method for the layout design of prestressed concrete structures is proposed. The Drucker–Prager yield criterion is used to predict the asymmetrical strength failure of concrete. The prestress is considered by making a reasonable assumption on the prestressing orientation in each element and adding an additional load vector to the structural equilibrium function. The proposed optimization model is thus formulated as to minimize the reinforcement material volume under Drucker–Prager yield constraints on elemental concrete local stresses. In order to give a reasonable definition of concrete local stress and prevent the stress singularity phenomenon, the local stress interpolation function and the ϵ -relaxation technique are adopted. The topology optimization problem is solved using the method of moving asymptotes combined with an active set strategy. Numerical examples are given to show the efficiency of the proposed optimization method in the layout design of prestressed concrete structures.

A multiplier active-set trust-region algorithm for solving constrained optimization problem

A new trust-region algorithm for solving a constrained optimization problem is introduced. In this algorithm, an active set strategy is used together with multiplier method to convert the computation of the trial step to easy trust-region subproblem similar to this for the unconstrained case. A convergence theory for this algorithm is presented. Under reasonable assumptions, it is shown that the algorithm is globally convergent. In particular, it is shown that, in the limit, a subsequence of the iteration sequence satisfies one of four types of stationary conditions. Namely, the infeasible Mayer–Bliss conditions, Fritz John’s conditions, the infeasible Fritz John’s conditions or KKT conditions.Preliminary numerical experiment on the algorithm is presented. The performance of the algorithm is reported. The numerical results show that our approach is of value and merit further investigation.

Path planning and collision avoidance for robots

An optimal control problem to find the fastest collision-free trajectory of a robot surrounded by obstacles is presented. The collision avoidance is based on linear programming arguments and expressed as state constraints. The optimal control problem is solved with a sequential programming method. In order to decrease the number of unknowns and constraints a backface culling active set strategy is added to the resolution technique.

A superlinearly convergent numerical algorithm for nonlinear programming

In this paper, a new algorithm is proposed to solve the nonlinear constrained optimization problems. Unlike sequential quadratic programming (SQP) type methods, this algorithm does not involve solutions of quadratic programs. It is merely necessary to solve systems of linear equations with small scale to obtain a direction. The scheme is based on an idea of ε-effective active set strategies. The theoretical analysis shows that global and superlinear convergence can be induced under some suitable conditions.