Gradient Projection Technique Research Articles

An approximating pseudospectral method with state‐dependent coefficient optimization for nonlinear optimal control problem

AbstractThe approximating sequence Riccati equation method is an efficient approach for solving the nonlinear optimal control problems, but its neglect of nonlinear dynamics and necessary optimality condition makes the control law difficult to satisfy the optimality. In this paper, an approximating pseudospectral method with state‐dependent coefficient optimization algorithm is proposed to solve this defect. By introducing the approximating pseudospectral method, the original nonlinear problem is transformed into a sequence of linear subproblems, which preserves the nonlinearity of solution. Then a state‐dependent coefficient optimization algorithm based on the gradient projection technique is proposed, which ensures the optimality of the control law. A double‐layer optimization structure is designed to facilitate the coordination between the approximating method and the optimization algorithm. Theoretical analysis proves the convergence of the proposed method. Comparative case studies illustrate the effectiveness in reducing the performance index and ensuring the optimality of the control law.

A New Projection Technique with Gradient Property to Solve Optimization Problems

In this study, a new gradient projection technique has been proposed that consists of three boundaries with achieving the unadulterated descent feature. In this technique, we worked on combining the conjugate gradient algorithm with projection techniques to obtain a new algorithm for solving a wide range of unconstrained optimization problems. We have established global convergence with some hypotheses, and it has become clear to us through our results that the new formula is good and promised.

Gradient methods exploiting spectral properties

We propose a new stepsize for the gradient method. It is shown that this new stepsize will converge to the reciprocal of the largest eigenvalue of the Hessian, when Dai-Yang's asymptotic optimal gradient method (Computational Optimization and Applications, 2006, 33(1): 73–88) is applied for minimizing quadratic objective functions. Based on this spectral property, we develop a monotone gradient method that takes a certain number of steps using the asymptotically optimal stepsize by Dai and Yang, and then follows by some short steps associated with this new stepsize. By employing one step retard of the asymptotic optimal stepsize, a nonmonotone variant of this method is also proposed. Under mild conditions, R-linear convergence of the proposed methods is established for minimizing quadratic functions. In addition, by combining gradient projection techniques and adaptive nonmonotone line search, we further extend those methods for general bound constrained optimization. Two variants of gradient projection methods combining with the Barzilai-Borwein stepsizes are also proposed. Our numerical experiments on both quadratic and bound constrained optimization indicate that the new proposed strategies and methods are very effective.

Dual feasible direction-finding nonlinear programming combined with metaheuristic approaches for exact overcurrent relay coordination

In the present paper, dual usable feasible direction-finding nonlinear programming associated with metaheuristic approaches is applied to directional overcurrent relay coordination. Rosen's gradient projection and Zoutendijk’s method are selected as nonlinear programming techniques, and the electro-search algorithm combined with Cuckoo optimization are chosen as the metaheuristic part. These algorithms are combined in a suitable manner based on their capabilities. First, all miscoordinations are removed by combining the metaheuristic part with a proposed simple manual manner. After eliminating all violations, the metaheuristic part is cooperated by Rosen's gradient projection technique. This loop is repeated until no change can be observed for a predefined number of iterations. Then, Zoutendijk’s method is applied to the results obtained by the previous algorithms. The proposed method is tested on both 8- and 30-bus test systems. Then, it is shown that the total primary operating times obtained, even for non-communication coordination scheme using relays with normally inverse characteristics, are considerably lower than those obtained by dual setting approaches.

Solution of Ill-Posed Nonconvex Optimization Problems with Accuracy Proportional to the Error in Input Data

The ill-posed problem of minimizing an approximately specified smooth nonconvex functional on a convex closed subset of a Hilbert space is considered. For the class of problems characterized by a feasible set with a nonempty interior and a smooth boundary, regularizing procedures are constructed that ensure an accuracy estimate proportional or close to the error in the input data. The procedures are generated by the classical Tikhonov scheme and a gradient projection technique. A necessary condition for the existence of procedures regularizing the class of optimization problems with a uniform accuracy estimate in the class is established.

Adaptive Inverse Control for Gripper Rotating System in Heavy-Duty Manipulators With Unknown Dead Zones

The gripper rotation system of a heavy-duty manipulator is generally driven by hydraulic motors, and it is difficult to guarantee the accuracy of the rotation angle using conventional control methods due to the existence of dead zones with unknown characteristics. This paper proposes an adaptive inverse dead-zone control method, in which the gradient projection technique is applied to design an adaptive updating law for estimating the inverse dead-zone parameters online. First, a dynamic model of the gripper rotation system is presented with an analysis of its dead-zone characteristics. Then, the controller is designed based on a Takagi–Sugeno fuzzy model, and the estimated values of the inverse dead-zone parameters are shown to converge to their true values. Finally, the performance of the proposed adaptive control method is experimentally demonstrated on an actual forging manipulator. Comparisons with a fixed inverse dead-zone control method demonstrate the effectiveness and applicability of the proposed control method.

A fast second-order parareal solver for fractional optimal control problems

The gradient projection technique has recently been used to solve the optimal control problems governed by a fractional diffusion equation. It lies in repeatedly solving the state and co-state equations derived from the optimality conditions, and the Crank–Nicolson (CN) scheme, which gives a second-order numerical solution, is a widely used method to solve these two equations. The goal of this paper is to implement the CN scheme in a parallel-in-time manner in the framework of the parareal algorithm. Because of the stiffness of the approximation matrix of the fractional operator, direct use of the CN scheme results in a convergence factor ρ satisfying as for the parareal algorithm, where denotes the space step-size. Here, we provide a new idea to let the parareal algorithm use the CN scheme as the basic component possessing a constant convergence factor , which is independent of . Numerical results are provided to show the efficiency of the proposed algorithm.

Time-Dependent Reliability Analysis Through Response Surface Method

In time-dependent reliability analysis, the first-passage method has been extensively used to evaluate structural reliability under time-variant service circumstances. To avoid computing the outcrossing rate in this method, surrogate modeling may provide an effective alternative for calculating the time-dependent reliability indices in structural analysis. A novel approach, namely time-dependent reliability analysis with response surface (TRARS), is thus introduced in this paper to estimate the time-dependent reliability for nondeterministic structures under stochastic loads. A Gaussian stochastic process is generated by using the expansion optimal linear estimation (EOLE) method which has proven to be more accurate and efficient than some series expansion discretization techniques. The random variables and maximum responses of uncertain structures are treated as the input and output parameters, respectively. Through introducing the response surface (RS) model, a novel iterative procedure is proposed in this study. A Bucher strategy is adopted to generate the initial sample points, and a gradient projection technique is used to generate new sampling points for updating the RS model in each iteration. The time-dependent reliability indices and probabilities of failure are thus obtained efficiently using the first-order reliability method (FORM) over a certain design lifetime. In this study, four demonstrative examples are provided for illustrating the accuracy and efficiency of the proposed method.

The calculation of the Stackelberg–Nash equilibrium as a fixed point problem in static hierarchical games

In this paper, a regularized version of the Extraproximal Method together with a Stochastic Approach is suggested to calculate the Stackelberg–Nash equilibrium in a N-person finite game. In this game, two levels of hierarchy in decision making are considered: one leader and (\(\hbox {N}-1\)) followers. Here, the followers playing according to the Nash equilibrium concept among themselves to the leaders announced strategy. An Extraproximal Technique is used to realize the application of a two-step procedure for finding the Nash equilibrium corresponding for the followers providing the loss function for a leader: at the first (or preliminary) step some predictive approximation of the a current approximation is calculated, at the second step (the main step of the iteration) this prediction is used to complete the current iteration. The next step, applying the stochastic gradient projection technique suggested, the Stackelberg–Nash equilibrium is found. The optimality conditions for a strategy being an equilibrium in participants game are derived based on the strong convexity of the \(\delta \)-regularized loss function. Simulation results illustrate the feasibility and the performance of this method.

Data-driven multi-stage multi-objective motion planning of mobile robots, application to near minimum power fuzzy parking

In this paper, the problem of multi-objective trajectory planning is studied for a wheeled mobile robot (WMR) in a crowded environment using a hybrid data-driven neuro-fuzzy system. This approach is composed of two basic stages involving a hard computing through a state space detailed modeling of the system and then a data driven evolutionary neuro-fuzzy system is used for online planning. Typically, a first pre-processing step involves an offline planning generating a large dataset of multi-objective trajectories, optimizing a time-power criterion while including robot, task, and workspace constraints. The discrete augmented Lagrangian is implemented on a decoupled form of the robot dynamics to solve for the resulting non-linear multi-objective optimal control problem. The final state constraint is satisfied with a gradient projection technique. The outcomes of this pre-processing step allow building a genetic neuro-fuzzy inference system to learn and capture the robot multi-objective dynamic behavior. Once this system is trained and optimized, it is used in a generalization phase to achieve online planning. The approach was also applied to the near optimal power fuzzy parking problem. Simulation results showing the effectiveness of the proposal are presented and discussed.

Physical total energy based objective function model for sparse reconstruction

Image reconstruction from sparse data is one of the key technologies in physical imaging, and it can often be mathematically described as an underdetermined linear inverse problem. Mathematical models for sparse reconstruction often choose the sparseness constraints or data fidelity term directly as objective functions. However, physics concepts and laws for these modeling or solving processes have never been explored. In this paper, sparse reconstruction is investigated for the first time from the perspective of physical motion. Firstly, a physical model is created to describe a particle motion in viscous medium, in which the particle gravity potential energy function is the norm of l2-l1 after the relaxation transformation. In discrete calculations, the particle displacement is determined by the corresponding iterative result, and its velocity can be described as the change between two adjacent iterations. Then, a new mathematical model based on the physical motion model is studied for sparse reconstruction, in which the total energy of particle is chosen as a new objective function and nonnegative displacements as constraints. This new model preserves sparse constrains and fidelity term of original l2-l1 model, and adds the constrains of deviations between two adjacent iterations so as to avoid oscillations caused by large deviations. Furthermore, a targeted gradient projection technique is adopted to solve such a reconstruction model, and its convergence is discussed as well. Especially in this algorithm, the gradient of this new objective function contains the iterative step of previous iteration, and such iterative steps play the role of physical inertia property in iterative process, which can effectively enlarge the iterative steps to accelerate the convergence and avoid local optima. Finally, two sets of experimental results are presented, including natural grayscale image reconstruction and micro focus X-ray defect detection in precision electronic package. The results demonstrate that the proposed method outperforms its competitors distinctly in time efficiency on the basis of guaranteeing the reconstruction quality. Additionally, on detecting internal defects in solder joint of integrated circuit, the proposed method is well performed in retaining edge details of the reconstructed micro focus X-ray images. Therefore, the proposed method can identify the solder joint internal defects more accurately and is more suitable to rapid and precise micro focus X-ray defect detection in industry.

A unified algorithm for mixed $$l_{2,p}$$ l 2 , p -minimizations and its application in feature selection

Recently, matrix norm $$l_{2,1}$$ l 2 , 1 has been widely applied to feature selection in many areas such as computer vision, pattern recognition, biological study and etc. As an extension of $$l_1$$ l 1 norm, $$l_{2,1}$$ l 2 , 1 matrix norm is often used to find jointly sparse solution. Actually, computational studies have showed that the solution of $$l_p$$ l p -minimization ( $$0<p<1$$ 0 < p < 1 ) is sparser than that of $$l_1$$ l 1 -minimization. The generalized $$l_{2,p}$$ l 2 , p -minimization ( $$p\in (0,1]$$ p ? ( 0 , 1 ] ) is naturally expected to have better sparsity than $$l_{2,1}$$ l 2 , 1 -minimization. This paper presents a type of models based on $$l_{2,p}\ (p\in (0, 1])$$ l 2 , p ( p ? ( 0 , 1 ] ) matrix norm which is non-convex and non-Lipschitz continuous optimization problem when $$p$$ p is fractional ( $$0<p<1$$ 0 < p < 1 ). For all $$p$$ p in $$(0, 1]$$ ( 0 , 1 ] , a unified algorithm is proposed to solve the $$l_{2,p}$$ l 2 , p -minimization and the convergence is also uniformly demonstrated. In the practical implementation of algorithm, a gradient projection technique is utilized to reduce the computational cost. Typically different $$l_{2,p}\ (p\in (0,1])$$ l 2 , p ( p ? ( 0 , 1 ] ) are applied to select features in computational biology.

A level set method for structural topology optimization with multi-constraints and multi-materials

Combining the vector level set model, the shape sensitivity analysis theory with the gradient projection technique, a level set method for topology optimization with multi-constraints and multi-materials is presented in this paper. The method implicitly describes structural material interfaces by the vector level set and achieves the optimal shape and topology through the continuous evolution of the material interfaces in the structure. In order to increase computational efficiency for a fast convergence, an appropriate nonlinear speed mapping is established in the tangential space of the active constraints. Meanwhile, in order to overcome the numerical instability of general topology optimization problems, the regularization with the mean curvature flow is utilized to maintain the interface smoothness during the optimization process. The numerical examples demonstrate that the approach possesses a good flexibility in handling topological changes and gives an interface representation in a high fidelity, compared with other methods based on explicit boundary variations in the literature.

Numerical analysis of leaving-face parameters in bound-constrained quadratic minimization*

In this work we focus our attention on the quadratic subproblem of trust-region algorithms for large-scale bound-constrained minimization. An approach that combines a mild active set strategy with gradient projection techniques is employed in the solution of large-scale bound-constrained quadratic problems. To fill in some gaps that have appeared in previous work, we propose and analyze heuristics which dynamically choose the parameters in charge of the decision of leaving or not the current face of the feasible set. The numerical analysis is based on problem from CUTE collection and randomly generated convex problems with controlled conditioning and degeneracy. The practical consequences of an appropriate decision of such parameters have shown to be crucial, particularly when dual degenerate problems are solved.

Null space velocity control with dynamically consistent pseudo-inverse

Null space velocity control is essential for achieving good behaviour of a redundant manipulator. Using the dynamically consistent pseudo-inverse, the task and null space motion and forces are decoupled. The paper presents a globally stable null space velocity controller and the gradient projection technique in conjunction with the dynamically consistent pseudo-inverse. The physical meaning and influence of the compensation terms in null the space velocity controller are explained. The performance of the proposed null space controller is tested on 4. d.o.f planar redundant manipulator interacting with the environment.

Influence of external forces on the behaviour of redundant manipulators

The paper considers the influence of external forces on the behaviour of a redundant manipulator. It is assumed that the forces can act anywhere on the body of the manipulator. First, the equivalent generalized forces in the task space and the null space are defined and several special manipulator configurations regarding the equivalent forces and torques are identified. Next, two measures for the quantification of the influence of external forces on the task space are proposed. These measures are then used in the control algorithm to minimize the influence of external forces on the task space position accuracy. The control is based on the redundancy resolution at the acceleration level and the gradient projection technique. Improvement of the position accuracy is illustrated using the simulation of a four link planar manipulator.

Coordinated-motion control of heavy-duty industrial machines with redundancy

SummaryAn implementation of a real-time scheme suitable for coordinated-motion control of a class of teleoperated industrial machines with redundancy is presented. An efficient gradient projection technique is adopted for the numerical solution. The scheme utilizes the redundancy to avoid the joint limits by minimizing a hyperbolic function of the joint distances from the mid-range. This new performance criterion is shown to be advantageous over similar criteria; both the joint-limit avoidance capability and the resulting joint velocity profiles can be adjusted by the appropriate choice of parameters introduced in the criterion. Previous criteria are shown to be special cases of the new criterion. Furthermore, the scheme includes a novel algorithm which incorporates the bounded joint velocities. The joint motions are determined considering both the required task in terms of the desired end-effector speed and the dynamic considerations, such as hydraulic circuit interdependency and power limitations.The feasibility and effectiveness of the implementation of the scheme is first tested through simulations of a Kaiser Spyder-like excavator machine on a PC-486 micro-computer. The robustness and real-time response of the scheme are then validated on a real-time excavator-based graphics simulator interfaced to a human operator through a joystick.

A new method for large-scale box constrained convex quadratic minimization problems

In this paper, we present a new method for minimizing a convex quadratic function of many variables with box constraints. The new algorithm is a modification of a method introduced recently by Friedlander and Martinez {SIAM J. on Optimization, February 1994). Following the lines of More and Toraldo (SIAM J. on Optimization 1, pp. 93-113), it combines an efficient unconstrained method with gradient projection techniques. The strategy for “leaving the current face” makes it possible to obtain convergence even when the Hessian is singular. Dual nondegeneracy is not assumed anywhere. The unconstrained minimization algorithm used within the faces was introduced by Barzilai and Borwein and analyzed by Raydan (IMA Journal of Numerical Analysis13, pp. 321-326)

On the Maximization of a Concave Quadratic Function with Box Constraints

A new method for maximizing a concave quadratic function with bounds on the variables is introduced. The new algorithm combines conjugate gradients with gradient projection techniques, as the algorithm of More and Toraldo [SIAM J. Optimization, 1 (1991), pp. 93–113] and other well-known methods do. A new strategy for the decision of leaving the current face is introduced that makes it possible to obtain finite convergence even for a singular Hessian and in the presence of dual degeneracy. Numerical experiments are presented.

Hypersphere Mapper: A Nonlinear Programming Approach to the Hypercube Embedding Problem

A nonlinear programming approach is introduced for solving the hypercube embedding problem. The basic idea of the proposed approach is to approximate the discrete space of an n-dimensional hypercube, i.e., { z: z ∈ {0, 1} n }, with the continuous space of an n-dimensional hypersphere, i.e., { x: x ∈ R n & || x|| 2 = 1}. The mapping problem is initially solved in the continuous domain by employing the gradient projection technique to a continuously differentiable objective function. The optimal process "locations" from the solution of the continuous hypersphere mapping problem are then discretized onto the n-dimensional hypercube. The proposed approach can solve, directly, the problem of mapping P processes onto N nodes for the general case where P > N. In contrast, competing embedding heuristics from the literature can produce only one-to-one mappings and cannot, therefore, be directly applied when P > N.