Penalty Problem Research Articles

A New Phase Current Reconstruction Scheme for Four-Phase SRM Drives Using Improved Converter Topology Without Voltage Penalty

Accurate current detection is of crucial importance for the operation and fault diagnosis of motor drives. In this paper, a new phase current reconstruction scheme is proposed for four-phase switched reluctance motors (SRMs) by using one dc-link current sensor without voltage penalty. First, an improved converter topology is proposed, which has fewer electronic components and a more compact structure compared with the conventional asymmetrical half-bridge converter. Then, an advanced pulse injection technology is developed to obtain the phase currents. By using the proposed approach, the cost and volume of the motor drive are reduced and the reliability of the system is improved. Most importantly, the scheme proposes a promising solution to the voltage penalty problem caused by the other existing strategies, which significantly increases the sampling accuracy without the restriction of the duty cycle of injected pulse. In order to verify the effectiveness of the proposed method, three classical control schemes including angular position control, current chopping control, and voltage pulse-width modulation control are all performed in the closed-loop system. The simulation and experiments based on a 150-W four-phase 8/6 SRM are carried out to validate the feasibility of this proposed scheme.

The fictitious domain method with L2‐penalty for the Stokes problem with the Dirichlet boundary condition

We consider the fictitious domain method with L2‐penalty for the Stokes problem with the Dirichlet boundary condition. First, we investigate the error estimates for the penalty method at the continuous level. We obtain the convergence of order in H1‐norm for the velocity and in L2‐norm for the pressure, where is the penalty parameter. The L2‐norm error estimate for the velocity is upgraded to . Moreover, we derive the a priori estimates depending on for the solution of the penalty problem. Next, we apply the finite element approximation to the penalty problem using the P1/P1 element with stabilization. For the discrete penalty problem, we prove the error estimate in H1‐norm for the velocity and in L2‐norm for the pressure, where h denotes the discretization parameter. For the velocity in L2‐norm, the convergence rate is improved to . The theoretical results are verified by the numerical experiments.

The fictitious domain method with H1-penalty for the Stokes problem with Dirichlet boundary condition

We consider the fictitious domain method with H1-penalty for the Stokes problem with Dirichlet boundary condition. First, for the continuous penalty problem, we obtain the optimal error estimate O(ϵ) for both the velocity and pressure, where ϵ is the penalty parameter. Moreover, we investigate the Hm-regularity for the solution of the penalty problem. Then, we apply the finite element method with the P1/P1 element to the penalty problem. Since the solution to the penalty problem has a jump in the traction vector, we introduce some interpolation/projection operators, as well as an inf-sup condition with the norm depending on ϵ. With the help of these preliminaries, we derive the error estimates for the finite element approximation. The theoretical results are verified by the numerical experiments.

METHAPHYSICS OF DEATH PENALTY

Purpose. The paper studies the problem of death penalty justifiableness in terms of democratic society from the metaphysical viewpoint. Philosophical argumentation to justify death penalty is proposed as opposed to the common idea of inhuman and uncivilized nature of court practice of sentencing to death. The essence of the study is not to rehabilitate law-based murder but to explain dialectic relation of the degrees of moral responsibility of criminals and society nourishing evildoers. The author believes that refusal from death penalty under the pretence of rule of humanism is just a liberal façade, plausible excuse for defective moral state of the society which, rejecting its own guiltiness share as for current disregards of the law, does not grow but downgrades proper human dignity. Methodology. The author applies an approach of dialectic reflection being guided by the perception of unity, relativeness and complementarity of evil and good striving to determine efficient way of resolving their contradictions in the context of moral progress of the society. Originality. Proposing philosophic approach to a death penalty problem instead of legal one, the author is not going to discuss the role of horrification, control or cruelty of the measure of restraint; moreover, he does not consider the issue of its efficiency or inefficiency. The author also does not concern vexation of mind of a criminal sentenced to life imprisonment for “humanitarian” reasons. The purpose of the author is to demonstrate that aim of the punishment is to achieve justice which becomes spiritual challenge and moral recompense not only for the criminal but for the whole society. Conclusions. Crime is first of all a problem of a society; thus, criminal behaviour of certain individuals should only be considered through a prism of moral state of the whole community. Attitude to a death penalty is the problem of spirituality and its dramatic sophistication. The author thinks that moral standards exclude any sentimental interpretation of humanism and mercy. Humanism is the imperative requiring both personal and social responsibility for the things humiliated in a victim and human dignity downtrodden in a criminal. Law-breaker cannot be liquidated without judicial safeguards as a dangerous animal. Severe punishment of the society adequately compensates a murderer his/her human dignity. As the society is guilty partially in the existing criminality, it accepts the fact of cruel punishment applied to redress an injustice.

The fictitious domain method for the Stokes problem with Neumann/free-traction boundary condition

We consider the fictitious domain method for the Stokes problem with Neumann/free-traction boundary condition. For the penalty method, we obtain the error estimate of order $$O(\epsilon )$$ , where $$\epsilon $$ is the penalty parameter. Next, we apply the finite element method to the penalty problem using the P1/P1 element with stabilization, and obtain the error estimate of discretization. The theoretical results are verified by the numerical experiments.

A New Smoothing Nonlinear Penalty Function for Constrained Optimization

In this study, a new smoothing nonlinear penalty function for constrained optimization problems is presented. It is proved that the optimal solution of the smoothed penalty problem is an approximate optimal solution of the original problem. Based on the smoothed penalty function, we develop an algorithm for finding an optimal solution of the optimization problems with inequality constraints. We further discuss the convergence of this algorithm and test this algorithm with three numerical examples. The numerical examples show that the proposed algorithm is feasible and effective for solving some nonlinear constrained optimization problems.

On optimal solutions of the constrained ℓ0 regularization and its penalty problem

The constrained regularization plays an important role in sparse reconstruction. A widely used approach for solving this problem is the penalty method, of which the least square penalty problem is a special case. However, the connections between global minimizers of the constrained problem and its penalty problem have never been studied in a systematic way. This work provides a comprehensive investigation on optimal solutions of these two problems and their connections. We give detailed descriptions of optimal solutions of the two problems, including existence, stability with respect to the parameter, cardinality and strictness. In particular, we find that the optimal solution set of the penalty problem is piecewise constant with respect to the penalty parameter. Then we analyze in-depth the relationship between optimal solutions of the two problems. It is shown that, in the noisy case the least square penalty problem probably has no common optimal solutions with the constrained problem for any penalty parameter. Under a mild condition on the penalty function, we establish that the penalty problem has the same optimal solution set as the constrained problem when the penalty parameter is sufficiently large. Based on the conditions, we further propose exact penalty problems for the constrained problem. Finally, we present a numerical example to illustrate our main theoretical results.

Multistage Convex Relaxation Approach to Rank Regularized Minimization Problems Based on Equivalent Mathematical Program with a Generalized Complementarity Constraint

This paper concerns itself with the rank regularized minimization problem with a spectral norm ball constraint. We reformulate this NP-hard problem as an equivalent MPGCC (mathematical program with a generalized complementarity constraint) and show that the penalty problem, yielded by moving the generalized complementarity constraint to the objective, is exact in the sense that it has the same global optimal solution set as the MPGCC does when the penalty parameter is over a threshold. Then, by solving the exact penalty problem in an alternating way, we propose a multistage convex relaxation approach, and provide its theoretical guarantee for the rank regularized least squares problem with a spectral norm ball constraint. Specifically, we derive the error and approximate rank bounds for the optimal solution of each stage and quantify the reduction of the error and approximate rank bounds of the first stage convex relaxation (which is exactly the nuclear norm relaxation) in the subsequent stages. In partic...

Computing minimum norm solution of linear systems of equations by the generalized Newton method

The aim of this paper is to find the minimum norm solution of a linear system of equations. The proposed method is based on presenting a view of solution on the dual exterior penalty problem of primal quadratic programming. To solve the unconstrained minimization problem, the generalized Newton method was employed and to guarantee its finite global convergence, the Armijo step size regulation was adopted. This method was tested on all systems selected in NETLIB 1 . Numerical results were compared with the MOSEK Optimization Software 2 on linear systems in NETLIB ( Table 1 ) and on linear systems generated by the Linear systems generator ( Table 2 ). 1 www.netlib.org 2 www.mosek.com

Linear programming twin support vector regression

In this paper, a new linear programming formulation of a 1-norm twin support vector regression is proposed whose solution is obtained by solving a pair of dual exterior penalty problems as unconstrained minimization problems using Newton method. The idea of our formulation is to reformulate TSVR as a strongly convex problem by incorporated regularization technique and then derive a new 1-norm linear programming formulation for TSVR to improve robustness and sparsity. Our approach has the advantage that a pair of matrix equation of order equals to the number of input examples is solved at each iteration of the algorithm. The algorithm converges from any starting point and can be easily implemented in MATLAB without using any optimization packages. The efficiency of the proposed method is demonstrated by experimental results on a number of interesting synthetic and real-world datasets.

A penalty-based scaling algorithm for the multi-period multi-product distribution planning problem

ABSTRACTMulti-period multi-product distribution planning problems are depicted as multi-commodity network flow problems where parameters may change over time. The corresponding mathematical formulation is presented for a discrete time setting, and it can also be used as an approximation for a continuous time setting. A penalty-based method which employs a cost-scaling approach is developed to solve some auxiliary penalty problems aiming to obtain an optimal solution for the original problem. The experiments on both random instances and case study problems show that the algorithm finds good-quality solutions with reasonable computational effort.

Smoothing of the lower-order exact penalty function for inequality constrained optimization

In this paper, we propose a method to smooth the general lower-order exact penalty function for inequality constrained optimization. We prove that an approximation global solution of the original problem can be obtained by searching a global solution of the smoothed penalty problem. We develop an algorithm based on the smoothed penalty function. It is shown that the algorithm is convergent under some mild conditions. The efficiency of the algorithm is illustrated with some numerical examples.

A sparse method for least squares twin support vector regression

Recently, some nonparallel plane regressors, such as twin support vector regression (TSVR), and least squares TSVR (LSTSVR), have been proposed and have attracted much attention. However, these algorithms are not sparse, which would make their learning speed low. In this paper, we propose a novel nonparallel plane regressor, which can automatically select the relevant features. Firstly, we introduce a regularization term to the objective function of LSTSVR, which can guarantee two quadratic programming problems (QPPs) are strong convex, implying that the proposed algorithm can obtain the global but unique solution. Secondly, the primal formulation is converted to a linear programming (LP) problem. Then, we solve the dual of the LP formulation by minimizing its exterior penalty problem, which would make our method yield very sparse solutions. In other words, this method can suppress input features so that it can obtain comparable regression performance when using fewer computational time. Numerical experiments on artificial dataset and benchmark datasets demonstrate the feasibility and validity of the proposed algorithm.

Unilateral problem for the Stokes equations: The well-posedness and finite element approximation

We consider the stationary Stokes equations under a unilateral boundary condition of Signorini's type, which is one of artificial boundary conditions in flow problems. Well-posedness is discussed through its variational inequality formulation. We also consider the finite element approximation for a regularized penalty problem. The well-posedness, stability and error estimates of optimal order are established. The lack of a coupled Babuška and Brezzi's condition makes analysis difficult. We offer a new method of analysis. Particularly, our device to treat the pressure is novel and of some interest. Numerical examples are presented to validate our theoretical results.

The stable duality of DC programs for composite convex functions

In this paper, we consider a composite DC optimization problem with a cone-convex system in locally convex Hausdorff topological vector spaces. By using the properties of the epigraph of the conjugate functions, some necessary and sufficient conditions which characterize the strong Fenchel-Lagrange duality and the stable strong Fenchel-Lagrange duality are given. We apply the results obtained to study the minmax optimization problem and $l_1$ penalty problem.

Smoothing Approximation to the Square-Root Exact Penalty Function

AbstractIn this paper, smoothing approximation to the square-root exact penalty functions is devised for inequality constrained optimization. It is shown that an approximately optimal solution of the smoothed penalty problem is an approximately optimal solution of the original problem. An algorithm based on the new smoothed penalty functions is proposed and shown to be convergent under mild conditions. Three numerical examples show that the algorithm is efficient.

Firefly penalty-based algorithm for bound constrained mixed-integer nonlinear programming

In this article, we aim to extend the firefly algorithm (FA) to solve bound constrained mixed-integer nonlinear programming (MINLP) problems. An exact penalty continuous formulation of the MINLP problem is used. The continuous penalty problem comes out by relaxing the integrality constraints and by adding a penalty term to the objective function that aims to penalize integrality constraint violation. Two penalty terms are proposed, one is based on the hyperbolic tangent function and the other on the inverse hyperbolic sine function. We prove that both penalties can be used to define the continuous penalty problem, in the sense that it is equivalent to the MINLP problem. The solutions of the penalty problem are obtained using a variant of the metaheuristic FA for global optimization. Numerical experiments are given on a set of benchmark problems aiming to analyze the quality of the obtained solutions and the convergence speed. We show that the firefly penalty-based algorithm compares favourably with the penalty algorithm when the deterministic DIRECT or the simulated annealing solvers are invoked, in terms of convergence speed.

Penalty Method for the Stationary Navier–Stokes Problems Under the Slip Boundary Condition

We consider the penalty method for the stationary Navier---Stokes equations with the slip boundary condition. The well-posedness and the regularity theorem of the penalty problem are investigated, and we obtain the optimal error estimate $$O(\epsilon )$$O(∈) in $$H^k$$Hk-norm, where $$\epsilon $$∈ is the penalty parameter. We are concerned with the finite element approximation with the P1b / P1 element to the penalty problem. The well-posedness of discrete problem is proved. We obtain the error estimate $$O(h+\sqrt{\epsilon }+h/\sqrt{\epsilon })$$O(h+∈+h/∈) for the non-reduced-integration scheme with $$d=2,3$$d=2,3, and the reduced-integration scheme with $$d=3$$d=3, where h is the discretization parameter and d is the spatial dimension. For the reduced-integration scheme with $$d=2$$d=2, we prove the convergence order $$O(h+\sqrt{\epsilon }+h^2/\sqrt{\epsilon })$$O(h+∈+h2/∈). The theoretical results are verified by numerical experiments.

One norm linear programming support vector regression

In this paper, a new linear programming formulation of a 1-norm support vector regression (SVR) is proposed whose solution is obtained by solving an exterior penalty problem in the dual space as an unconstrained minimization problem using Newton method. The solution of modified unconstrained minimization problem reduces to solving just system of linear equations as opposed to solving quadratic programming problem in SVR, which leads to extremely simple and fast algorithm. The algorithm converges from any starting point and can be easily implemented in MATLAB without using any optimization packages. The main advantage of the proposed approach is that it leads to a robust and sparse model representation meaning that many components of the optimal solution vector will become zero and therefore the decision function can be determined using much less number of support vectors in comparison to SVR, smooth SVR (SSVR) and weighted SVR (WSVR). To demonstrate its effectiveness, experiments were performed on well-known synthetic and real-world benchmark datasets. Similar or better generalization performance of the proposed method in less training time in comparison with SVR, SSVR and WSVR clearly exhibits its suitability and applicability.

Regression and classification using extreme learning machine based on L1-norm and L2-norm

Extreme learning machine (ELM) is a very simple machine learning algorithm and it can achieve a good generalization performance with extremely fast speed. Therefore it has practical significance for data analysis in real-world applications. However, it is implemented normally under the empirical risk minimization scheme and it may tend to generate a large-scale and over-fitting model. In this paper, an ELM model based on L1-norm and L2-norm regularizations is proposed to handle regression and multiple-class classification problems in a unified framework. The proposed model called L1–L2-ELM combines the grouping effect benefits of L2 penalty and the tendency towards sparse solution of L1 penalty, thus it can control the complexity of the network and prevent over-fitting. To solve the mixed penalty problem, the separate elastic net algorithm and Bayesian information criterion (BIC) are adopted to find the optimal model for each response variable. We test the L1–L2-ELM algorithm on one artificial case and nine benchmark data sets to evaluate its performance. Simulation results have shown that the proposed algorithms outperform the original ELM as well as other advanced ELM algorithms in terms of prediction accuracy, and it is more robust in both regression and classification applications.