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In this paper, an active set recognition technique is suggested, and then a modified nonmonotonic line search rule is presented to enhance the efficiency of the nonmonotonic line search rule, in which we introduce a new parameter formula to attempt to control the nonmonotonic degree of the line search, and thus improve the chance of discovering the global minimum. By using a modified linear search and an active set recognition technique, a global convergence gradient solution for nonnegative matrix factorization (NMF) based on an alternating nonnegative least squares framework is proposed. We used a Barzilai-Borwein step size and greater step-size tactics to speed up the convergence. Finally, a large number of numerical experiments were carried out on synthetic and image datasets, and the results showed that our presented method was effective in calculating the speed and solution quality. [ABSTRACT FROM AUTHOR]

Optimal power flow (OPF) is a crucial tool in the operation and planning of modern power systems. However, as power system optimization shifts towards larger-scale frameworks, and with the growing integration of distributed generations, the computational time and memory requirements of solving the alternating current (AC) OPF problems can increase exponentially with system size, posing computational challenges. In recent years, machine learning (ML) has demonstrated notable advantages in efficient computation and has been extensively applied to tackle OPF challenges. This paper presents five commonly employed OPF transformation techniques that leverage ML, offering a critical overview of the latest applications of advanced ML in solving OPF problems. The future directions in the application of machine learning to AC OPF are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we first develop an active set identification technique, and then we suggest a modified nonmonotone line search rule, in which a new parameter formula is introduced to control the degree of the nonmonotonicity of line search. By using the modified line search and the active set identification technique, we propose a global convergent method to solve the NMF based on the alternating nonnegative least squares framework. In addition, the larger step size technique is exploited to accelerate convergence. Finally, a large number of numerical experiments are carried out on synthetic and image datasets, and the results show that our presented method is effective in calculating speed and solution quality. [ABSTRACT FROM AUTHOR]

In this paper, an active-set strategy is used together with a penalty method and a trust-region technique to solve a warehouses location problem. The trust region is used to modify the local method in such a way that it is guaranteed to converge at all even if the starting point is far away from the solution. The trust-region method is a well-accepted technique in nonlinear optimization to assure global convergence and is more robust when they deal with rounding errors. One of the advantages of trust-region method is that it does not require the objective function of the model to be convex. The warehouses location problem involves the determination of the number and size of service center (warehouses) to supply a set of demand centers so as to minimize total distribution cost. The proposed approach is tested on two problems to confirm the effectiveness of the algorithm. Our results with the proposed approach have been compared to those reported in the literature. [ABSTRACT FROM AUTHOR]

In this paper, a reduced interior-point (RIP) algorithm is introduced to generate a Pareto optimal front for multi-objective constrained optimization (MOCP) problem. A weighted Tchebychev metric approach is used together with achievement scalarizing function approach to convert MOCP problem to a single-objective constrained optimization (SOCO) problem. An active-set technique is used together with a Coleman-Li scaling matrix and a decrease interior-point method to solve SOCO problem. A Matlab implementation of RIP algorithm was used to solve three cases and application. The results showed that the RIP algorithm is promising when compared with well-known algorithms and the computations may be superior relevant for comprehending real-world application problems. [ABSTRACT FROM AUTHOR]

In this paper, an infeasible QP-free method without penalty function is proposed for inequality constrained optimization. We first compute a fundamental direction and then bend the search direction based on the constraint function and the Lagrange multiplier. Based on the modified nonmonotone filter technique, the acceptable criterion of trial points is relaxed and Maratos effects are avoided to a certain degree. At each iteration, only two or three systems of linear equations with the same coefficient are needed to solve to obtain the search direction. Under suitable conditions, the global convergence of the algorithm is proved without the strict complementarity conditions. In the end, some numerical results are reported. [ABSTRACT FROM AUTHOR]

In this paper, we develop an active set identification technique for the ℓ 0 regularization optimization. Such a technique has a strong ability to identify the zero components in a neighbourhood of a strict L-stationary point. Based on the identification technique, we propose an active set Barzilar–Borwein algorithm and prove that any limit point of the sequence generated by the algorithm is a strong stationary point. Some preliminary numerical results are provided, showing that the method is promising. [ABSTRACT FROM AUTHOR]

In this paper, a new algorithm for tackling nonlinear constrained multi-objective optimization problems is introduced. This algorithm, which we call relaxed the interactive sequential hybrid optimization technique (relaxed I-SHOT), is based on a hybrid method between the I-SHOT method and step method (STEM) to transform a multi-objective problem into a single-objective one. An active set strategy is used together with a suggested penalty method to transform a single-objective constrained optimization problem into unconstrained one. A trust-region globalization strategy is added to the algorithm to solve the obtained unconstrained problem to ensure global convergence. The suggested approach is utilized to solve the multi-objective economic emission load dispatch (EELD) problems to assert the effectiveness of the proposed algorithm. The proposed algorithm is tested on the standard IEEE 30-bus 6-generator test system. The results of the proposed approach are compared against those reported in the literature. The comparison asserts the potential of the proposed algorithm to solve the EELD problem. [ABSTRACT FROM AUTHOR]

Base kernel selection, the task of selecting multiple good kernels, is a key issue in multiple kernel learning (MKL) algorithms. This paper introduces a new framework for base strong kernel selection, named multiple random compact Gaussian kernel learning (MRKL), which is efficient and provides good off-the-shelf usability. It employs a random compact Gaussian kernel to generate a number of kernel candidates, which provide complementary information for investigating the data distribution. Then, it selects strong kernels based on a maximal-accuracy-minimal-difference criterion. It addresses the key challenge to MKL algorithms when each base kernel has a single weight and, as a result, different kernels cannot capture the data characteristics in different subspaces of the input space. MRKL is independent of the MKL algorithm, so that it can be combined with any existing MKL algorithm. As examples, we combine MRKL with two popular MKL algorithms, the same weighted-based-kernel combination algorithm and optimization method-based-kernel combination algorithm. We also design a fast active set algorithm for training support vector machine (SVM) with a combined kernel. Empirical results on benchmark datasets demonstrate that MRKL significantly improves existing MKL algorithms and outperforms other state-of-the-art MKL algorithms. • MRKL provides complementary information for investigating the data distribution. • We propose a new kernel selection criterion called MAMD. • MRKL can be combined with popular kernel combination strategies. [ABSTRACT FROM AUTHOR]

• The aim of demand-side management is to reduce peak consumption during peak hours. • Low participation of consumers is the vital challenge in DSM. • A flexible DSM method is presented in accordance with consumers' constraints. • The lowest and highest possible consumption of each consumer are considered. • The stability of consumers' total consumption in some time periods is considered. Increasing electrical energy consumption during peak hours leads to increased electrical energy losses and the spread of environmental pollution. For this reason, demand-side management programs have been introduced to reduce consumption during peak hours. The most important challenge of these programs is the low participation of consumers due to their special constraints. Hence, presenting a flexible demand-side management program that optimizes the consumption pattern of consumers in accordance with their requested constraints will have a significant impact on further participation of consumers in reducing the peak consumption. In this paper, a flexible demand-side management program is presented that provides the optimal consumption patterns, by using the stair-step pricing mechanism and considering the consumers requested constraints. The proposed method is carried out for a combination of consumers with various requested constraints such as the lowest and highest possible consumption per hour and the stability of the total consumption of each consumer in a certain period of time. The simulation outcome showed that the proposed algorithm is able to encourage consumers to participate more in energy management program by meeting their various requested constraints. Peak load total consumption experienced a decline of 25.09% from 36649.4 to 27454.3 kW-hours. As a result, the daily energy cost had been reduced by 10.27 % from 13854.466 to 12,431 dollars. This cost reduction was derived from reducing the need for additional energy generation during peak hours due to transferring consumptions to the lower-priced hours. [ABSTRACT FROM AUTHOR]

Pulse width modulation (PWM) technology is widely used in traction converters for high-speed railways. The harmonic distribution caused by PWM is quite extensive, and increases the possibility of grid-train coupling resonance in the traction power supply system (TPSS). This paper first analyzes the mechanism of resonance, when the characteristic harmonic frequency of a four-quadrant converter (4QC) current that injects into the traction grid matches the resonant frequency of the traction grid, which may result in resonance in the system. To suppress resonance, this paper adopts specific harmonic elimination-pulse width modulation (SHE-PWM) technology combined with a transient direct current control strategy to eliminate the harmonics in the resonant frequency, which may suppress the grid-train coupling resonance. Due to the fact that the SHE-PWM process with multiple switching angles contains complex transcendental equations, the initial value is difficult to provide, and is difficult to solve using ordinary iterative algorithms. In this paper, an active-set secondary optimization method is used to solve the equation. The algorithm has the benefits of low dependence on initial values, fast convergence and high solution accuracy. Finally, the feasibility of the resonant suppression algorithm is verified by means of Matlab simulation. [ABSTRACT FROM AUTHOR]

Optimal power flow (OPF) is a crucial tool in the operation and planning of modern power systems. However, as power system optimization shifts towards larger-scale frameworks, and with the growing integration of distributed generations, the computational time and memory requirements of solving the alternating current (AC) OPF problems can increase exponentially with system size, posing computational challenges. In recent years, machine learning (ML) has demonstrated notable advantages in efficient computation and has been extensively applied to tackle OPF challenges. This paper presents five commonly employed OPF transformation techniques that leverage ML, offering a critical overview of the latest applications of advanced ML in solving OPF problems. The future directions in the application of machine learning to AC OPF are also discussed.

In this paper, we study an algorithm (which we call the support reduction algorithm) that can be used to compute non-parametric M-estimators in mixture models. The algorithm is compared with natural competitors in the context of convex regression and the ‘Aspect problem’ in quantum physics. [ABSTRACT FROM AUTHOR]

In this paper, we first develop an active set identification technique, and then we suggest a modified nonmonotone line search rule, in which a new parameter formula is introduced to control the degree of the nonmonotonicity of line search. By using the modified line search and the active set identification technique, we propose a global convergent method to solve the NMF based on the alternating nonnegative least squares framework. In addition, the larger step size technique is exploited to accelerate convergence. Finally, a large number of numerical experiments are carried out on synthetic and image datasets, and the results show that our presented method is effective in calculating speed and solution quality.

Constructing active sets is a key part of the Multivariate Decomposition Method. An algorithm for constructing optimal or quasi-optimal active sets is proposed in the paper. By numerical experiments, it is shown that the new method can provide sets that are significantly smaller than the sets constructed by the already existing method. The experiments also show that the superposition dimension could surprisingly be very small, at most 3, when the error demand is not smaller than 1 0 − 3 and the weights decay sufficiently fast. [ABSTRACT FROM AUTHOR]

This paper presents a new approach for an online solution of the inverse kinematics problem based on nonlinear optimization for robots with joint physical constraints. The inverse kinematics problem is stated as a constrained nonlinear optimization problem and is solved using Kuhn{Tucker conditions analysis. The nonlinear multivariable optimization problem is locally converted into independent local linear constrained subproblems and each subproblem is analytically solved. For each joint, position limits and velocity limits are considered as physical constraints. The proposed structure benefits from a very low complexity design. The proposed method is fast and requires few calculations. The convergence of the proposed algorithm is proven based on the Lyapunov function. While keeping the algorithm stable, it can navigate the manipulator to a desired position under joint physical limits. The algorithm is simulated on a 7-DOF PA-10 manipulator. Results indicate good efficiency for the proposed structure in constrained path planning of joint space. [ABSTRACT FROM AUTHOR]

In 1974 Lawson and Hanson produced a seminal active set strategy to solve least-squares problems with non-negativity constraints that remains popular today. In this paper we present TNT-NN, a new active set method for solving non-negative least squares (NNLS) problems. TNT-NN uses a different strategy not only for the construction of the active set but also for the solution of the unconstrained least squares sub-problem. This results in dramatically improved performance over traditional active set NNLS solvers, including the Lawson and Hanson NNLS algorithm and the Fast NNLS (FNNLS) algorithm, allowing for computational investigations of new types of scientific and engineering problems. For the small systems tested (5000 × 5000 or smaller), it is shown that TNT-NN is up to 95 × faster than FNNLS. Recent studies in rock magnetism have revealed a need for fast NNLS algorithms to address large problems (on the order of 10 5 × 10 5 or larger). We apply the TNT-NN algorithm to a representative rock magnetism inversion problem where it is 60× faster than FNNLS. We also show that TNT-NN is capable of solving large (45000 × 45000) problems more than 150 × faster than FNNLS. These large test problems were previously considered to be unsolvable, due to the excessive execution time required by traditional methods. [ABSTRACT FROM AUTHOR]

A new exchange method is presented for semi-infinite optimization problems with polyhedron constraints. The basic idea is to use an active set strategy as exchange rule to construct an approximate problem with finitely many constraints at each iteration. Under mild conditions, we prove that the proposed algorithm terminates in a finite number of iterations and guarantees that the solution of the resulting approximate problem at final iteration converges to the solution of the original problem within arbitrarily given tolerance. Numerical results indicate that the proposed algorithm is efficient and promising. [ABSTRACT FROM AUTHOR]

• Modifier adaptation for real-time optimization requires estimating plant gradients. • Efficient ways to implement estimation of plant gradients in practice are proposed. • One approach uses the knowledge of the active set to reduce the number of modifiers. • Another approach determines the dominant gradients using sensitivity analysis. • Near plant optimality is reached efficiently and the experimental effort is reduced. In real-time optimization, the solution quality depends on the model ability to predict the plant Karush–Kuhn–Tucker (KKT) conditions. In the case of non-parametric plant-model mismatch, one can add input-affine modifiers to the model cost and constraints as is done in modifier adaptation (MA). These modifiers require estimating the plant cost and constraint gradients. This paper discusses two ways of reducing the number of input directions, thereby improving the efficiency of MA in practice. The first approach capitalizes on the knowledge of the active set to reduce the number of KKT conditions. The second approach determines the dominant gradients using sensitivity analysis. This way, MA reaches near plant optimality efficiently by adapting the first-order modifiers only along the dominant input directions. These approaches allow generating several alternative MA schemes, which are analyzed in terms of the number of degrees of freedom and compared in a simulated study of the Williams–Otto plant. [ABSTRACT FROM AUTHOR]

In this paper, an active set quasi-Newton method for bound constrained nonlinear equation is proposed. By using this active set technique, we only need to solve a reduced dimension linear equation at each iteration to generate the search direction. The algorithm is a combination of the quasi-Newton method and projection method. Firstly we use the quasi-Newton step as the trial step and then use a projection technique to generate the next iteration point. Our key observation is that the algorithm generates a bounded iteration sequence automatically even if the bounds are equal to infinity and the global convergence is obtained in the sense that the whole sequence converges to the stationary point. The numerical tests show the efficiency of the algorithm.

Abstract: Sequential quadratic programming (SQP) has been one of the most important methods for solving nonlinearly constrained optimization problems. In this paper, we present and study an active set SQP algorithm for inequality constrained optimization. The active set technique is introduced which results in the size reduction of quadratic programming (QP) subproblems. The algorithm is proved to be globally convergent. Thus, the results show that the global convergence of SQP is still guaranteed by deleting some “redundant” constraints. [Copyright &y& Elsevier]

In this paper, a new algorithm for tackling nonlinear constrained multi-objective optimization problems is introduced. This algorithm, which we call relaxed the interactive sequential hybrid optimization technique (relaxed I-SHOT), is based on a hybrid method between the I-SHOT method and step method (STEM) to transform a multi-objective problem into a single-objective one. An active set strategy is used together with a suggested penalty method to transform a single-objective constrained optimization problem into unconstrained one. A trust-region globalization strategy is added to the algorithm to solve the obtained unconstrained problem to ensure global convergence. The suggested approach is utilized to solve the multi-objective economic emission load dispatch (EELD) problems to assert the effectiveness of the proposed algorithm. The proposed algorithm is tested on the standard IEEE 30-bus 6-generator test system. The results of the proposed approach are compared against those reported in the literature. The comparison asserts the potential of the proposed algorithm to solve the EELD problem.

In this paper, a reduced interior-point (RIP) algorithm is introduced to generate a Pareto optimal front for multi-objective constrained optimization (MOCP) problem. A weighted Tchebychev metric approach is used together with achievement scalarizing function approach to convert MOCP problem to a single-objective constrained optimization (SOCO) problem. An active-set technique is used together with a Coleman–Li scaling matrix and a decrease interior-point method to solve SOCO problem. A Matlab implementation of RIP algorithm was used to solve three cases and application. The results showed that the RIP algorithm is promising when compared with well-known algorithms and the computations may be superior relevant for comprehending real-world application problems.

• From 8 up to 78 pct decrease in the number of Hessian multiplication compared with standard expansion. • Two new adaptive step lengths for expansion step with 5 different combination of vectors used for their computations and search directions. • New alternative expansion strategy based on projected conjugate gradient step. The paper investigates strategies for expansion of active set that can be employed by the MPRGP algorithm. The standard MPRGP expansion uses a projected line search in the free gradient direction with a fixed step length. Such a scheme is often too slow to identify the active set, requiring a large number of expansions. We propose to use adaptive step lengths based on the current gradient, which guarantees the decrease of the unconstrained cost function with different gradient-based search directions. Moreover, we also propose expanding the active set by projecting the optimal step for the unconstrained minimization. Numerical experiments demonstrate the benefits (up to 78% decrease in the number of Hessian multiplications) of our expansion step modifications on two benchmarks – contact problem of linear elasticity solved by TFETI and machine learning problems of SVM type, both implemented in PERMON toolbox. [ABSTRACT FROM AUTHOR]

Quadratic optimization problems (QPs) are ubiquitous, and solution algorithms have matured to a reliable technology. However, the precision of solutions is usually limited due to the underlying floating-point operations. This may cause inconveniences when solutions are used for rigorous reasoning. We contribute on three levels to overcome this issue. First, we present a novel refinement algorithm to solve QPs to arbitrary precision. It iteratively solves refined QPs, assuming a floating-point QP solver oracle. We prove linear convergence of residuals and primal errors. Second, we provide an efficient implementation, based on SoPlex and qpOASES that is publicly available in source code. Third, we give precise reference solutions for the Maros and Mészáros benchmark library. [ABSTRACT FROM AUTHOR]

In this paper, an active-set strategy is used together with a penalty method and a trust-region technique to solve a warehouses location problem. The trust region is used to modify the local method in such a way that it is guaranteed to converge at all even if the starting point is far away from the solution. The trust-region method is a well-accepted technique in nonlinear optimization to assure global convergence and is more robust when they deal with rounding errors. One of the advantages of trust-region method is that it does not require the objective function of the model to be convex. The warehouses location problem involves the determination of the number and size of service center (warehouses) to supply a set of demand centers so as to minimize total distribution cost. The proposed approach is tested on two problems to confirm the effectiveness of the algorithm. Our results with the proposed approach have been compared to those reported in the literature.

In this paper we give a general framework for isotone optimization. First we discuss a generalized version of the pool-adjacent-violators algorithm (PAVA) to minimize a separable convex function with simple chain constraints. Besides of general convex functions we extend existing PAVA implementations in terms of observation weights, approaches for tie handling, and responses from repeated measurement designs. Since isotone optimization problems can be formulated as convex programming problems with linear constraints we the develop a primal active set method to solve such problem. This methodology is applied on specific loss functions relevant in statistics. Both approaches are implemented in the R package isotone.

new algorithm for solving large scale bound constrained minimization problems is proposed. The algorithm is based on an accurate identification technique of the active set proposed by Facchinei, Fischer and Kanzow in 1998. A further division of the active set yields the global convergence of the new algorithm. In particular, the convergence rate is superlinear without requiring the strict complementarity assumption. Numerical tests demonstrate the efficiency and performance of the present strategy and its comparison with some existing active set strategies. [ABSTRACT FROM AUTHOR]

We employ the active set strategy which was proposed by Facchinei for solving large scale bound constrained optimization problems. As the special structure of the bound constrained problem, a simple rule is used for updating the multipliers. Numerical results show that the active set identification strategy is practical and efficient. [ABSTRACT FROM AUTHOR]