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The stripe noise existing in remote sensing images badly degrades the visual quality and restricts the precision of data analysis. Therefore, many destriping models have been proposed in recent years. In contrast to these existing models, in this paper, we propose a nonconvex model with a DC function (i.e., the difference of convex functions) structure to remove the strip noise. To solve this model, we make use of the DC structure and apply an inexact proximal majorization-minimization algorithm with each inner subproblem solved by the alternating direction method of multipliers. It deserves mentioning that we design an implementable stopping criterion for the inner subproblem, while the convergence can still be guaranteed. Numerical experiments demonstrate the superiority of the proposed model and algorithm. [ABSTRACT FROM AUTHOR]

This is the Preface to the Special Issue of JOTA dedicated to the 60th birthday of Professor Aram V. Arutyunov [ABSTRACT FROM AUTHOR]

This paper presents a new adaptive steplength for the gradient method, which exploits the advantages the two steplengths proposed by Barzilai and Borwein. Particularly, the proposed steplength is based on an optimal step size determined by minimizing a merit function constructed as a convex combination of the cost function and its gradient norm. The global convergence and some theoretical properties related to the proposed gradient method are provided. Finally, some computational studies are included to highlight the efficiency and effectiveness of the new approach. [ABSTRACT FROM AUTHOR]

The Universum provides prior knowledge about data in the mathematical problem to improve the generalization performance of the classifiers. Several works have shown that the Universum twin support vector machine (U -TSVM) is an efficient method for binary classification problems. In this paper, we improve the U -TSVM method and propose an improved Universum twin bounded support vector machine (named as IUTBSVM). Indeed, by introducing different Lagrangian functions for the primal problems, we obtain new dual formulations of U -TSVM so that we do not need to compute inverse matrices. To reduce the computational time of the proposed method, we suggest a smaller size of the rectangular kernel matrices than the other methods. Numerical experiments on gender classification of human faces, handwritten digits recognition, and several UCI benchmark data sets indicate that the IUTBSVM is more efficient than the other four algorithms, namely U -SVM, TSVM, U -TSVM, and IUTSVM in the sense of the classification accuracy. [ABSTRACT FROM AUTHOR]

The present paper shows how the linear independence constraint qualification (LICQ) can be combined with some conditions put on the first-order and second-order derivatives of the objective function and the constraint functions to ensure the Robinson stability and the Lipschitz-like property of the stationary point set map of a general C2-smooth parametric constrained optimization problem. So, a part of the results in two preceding papers of the authors [J. Optim. Theory Appl. 180 (2019), 91–116 (Part 1); 117–139 (Part 2)], which were obtained for a problem with just one inequality constraint, now has an adequate extension for problems having finitely many equality and inequality constraints. Our main tool is an estimate of B. S. Mordukhovich and R. T. Rockafellar [SIAM J. Optim. 22 (2012), 953–986; Theorem 3.3] for a second-order partial subdifferential of a composite function. The obtained results are illustrated by three examples. [ABSTRACT FROM AUTHOR]

The multi-objective integer programming problem often occurs in multi-criteria decision-making situations, where the decision variables are integers. In the present paper, we have discussed an algorithm for finding all efficient solutions of a multi-objective integer quadratic programming problem. The proposed algorithm is based on the aspect that efficient solutions of a multi-objective integer quadratic programming problem can be obtained by enumerating ranked solutions of an integer quadratic programming problem. For determining ranked solutions of an integer quadratic programming problem, we have constructed a related integer linear programming problem and from ranked solutions of this integer linear programming problem, ranked solutions of the original integer quadratic programming problem are generated. Theoretically, we have shown that the developed method generates the set of all efficient solutions in a finite number of steps, and numerically we have elaborated the working of our algorithm and compared our results with existing algorithms. Further, we have analyzed that the developed method is efficient for solving a multi-objective integer quadratic programming problem with a large number of constraints, variables and objectives. [ABSTRACT FROM AUTHOR]

This paper proposes a novel matrix rank-one decomposition for quaternion Hermitian matrices, which admits a stronger property than the previous results in Ai W et al (Math Progr 128(1):253–283, 2011), Huang Y, Zhang S (Math Oper Res 32(3):758–768, 2007), Sturm JF, Zhang S (Math Oper Res 28(2):246–267 2003). The enhanced property can be used to drive some improved results in joint numerical range, S -Procedure and quadratically constrained quadratic programming (QCQP) in the quaternion domain, demonstrating the capability of our new decomposition technique. [ABSTRACT FROM AUTHOR]

In this paper, we give an algorithm that finds an ϵ -approximate solution to a mixed integer quadratic programming (MIQP) problem. The algorithm runs in polynomial time if the rank of the quadratic function and the number of integer variables are fixed. The running time of the algorithm is expected unless P = NP. In order to design this algorithm we introduce the novel concepts of spherical form MIQP and of aligned vectors, and we provide a number of results of independent interest. In particular, we give a strongly polynomial algorithm to find a symmetric decomposition of a matrix, and show a related result on simultaneous diagonalization of matrices. [ABSTRACT FROM AUTHOR]

Linear inverse problems arise in diverse engineering fields especially in signal and image reconstruction. The development of computational methods for linear inverse problems with sparsity is one of the recent trends in this field. The so-called optimal k-thresholding is a newly introduced method for sparse optimization and linear inverse problems. Compared to other sparsity-aware algorithms, the advantage of optimal k-thresholding method lies in that it performs thresholding and error metric reduction simultaneously and thus works stably and robustly for solving medium-sized linear inverse problems. However, the runtime of this method is generally high when the size of the problem is large. The purpose of this paper is to propose an acceleration strategy for this method. Specifically, we propose a heavy-ball-based optimal k-thresholding algorithm and its relaxed variants for sparse linear inverse problems. The convergence of these algorithms is shown under the restricted isometry property. In addition, the numerical performance of the heavy-ball-based relaxed optimal k-thresholding pursuit (HBROTP) has been evaluated, and simulations indicate that HBROTP admits robustness for signal and image reconstruction even in noisy environments. [ABSTRACT FROM AUTHOR]

Tokamaks are the most promising devices to prove the feasibility of energy production using nuclear fusion on Earth which is foreseen as a possible source of energy for the next centuries. In large tokamaks with superconducting poloidal field (PF) coils, the problem of avoiding saturation of the currents is of paramount importance, especially for a reactor such as the European demonstration fusion power plant DEMO. Indeed, reaching the current limits during plasma operation may cause a loss of control of the plasma shape and/or current, leading to a major disruption. Therefore, a current limit avoidance (CLA) system is essential to assure safe operation. Three different algorithms to be implemented within a CLA system are proposed in this paper: two are based on online solutions of constrained optimization problems, while the third one relies on dynamic allocation. The performance assessment for all the proposed solutions is carried out by considering challenging operation scenarios for the DEMO reactor, such as the case where more than one PF current simultaneously saturates during the discharge. An evaluation of the computational burden needed to solve the allocation problem for the various proposed alternatives is also presented, which shows the compliance of the optimization-based approaches with the envisaged deadlines for real-time implementation of the DEMO plasma magnetic control system. [ABSTRACT FROM AUTHOR]

In this paper, we provide a finitely terminated yet efficient approach to compute the Euclidean projection onto the ordered weighted ℓ1 (OWL1) norm ball. In particular, an efficient semismooth Newton method is proposed for solving the dual of a reformulation of the original projection problem. Global and local quadratic convergence results, as well as the finite termination property, of the algorithm are proved. Numerical comparisons with the two best-known methods demonstrate the efficiency of our method. In addition, we derive the generalized Jacobian of the studied projector which, we believe, is crucial for the future designing of fast second-order nonsmooth methods for solving general OWL1 norm constrained problems. [ABSTRACT FROM AUTHOR]

Square-root Lasso problems have already be shown to be robust regression problems. Furthermore, square-root regression problems with structured sparsity also plays an important role in statistics and machine learning. In this paper, we focus on the numerical computation of large-scale linearly constrained sparse group square-root Lasso problems. In order to overcome the difficulty that there are two nonsmooth terms in the objective function, we propose a dual semismooth Newton (SSN) based augmented Lagrangian method (ALM) for it. That is, we apply the ALM to the dual problem with the subproblem solved by the SSN method. To apply the SSN method, the positive definiteness of the generalized Jacobian is very important. Hence we characterize the equivalence of its positive definiteness and the constraint nondegeneracy condition of the corresponding primal problem. In numerical implementation, we fully employ the second order sparsity so that the Newton direction can be efficiently obtained. Numerical experiments demonstrate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we develop a semidefinite relaxation-based branch-and-bound algorithm that exploits the chordal sparsity patterns of the max-cut problem. We first study how the chordal sparsity pattern affects the hardness of a max-cut problem. To do this, we derive a polyhedral relaxation based on the clique decomposition of the chordal sparsity patterns and prove some sufficient conditions for the tightness of this polyhedral relaxation. The theoretical results show that the max-cut problem is easy to solve when the sparsity pattern embedded in the problem has a small treewidth and the number of vertices in the intersection of maximal cliques is small. Based on the theoretical results, we propose a new branching rule called hierarchy branching rule, which utilizes the tree decomposition of the sparsity patterns. We also analyze how the proposed branching rule affects the chordal sparsity patterns embedded in the problem, and explain why it can be effective. The numerical experiments show that the proposed algorithm is superior to those known algorithms using classical branching rules and the state-of-the-art solver BiqCrunch on most instances with sparsity patterns arisen in practical applications. [ABSTRACT FROM AUTHOR]

The quadratically constrained quadratic programming problem often appears in various fields such as engineering practice, management science and network communication. This paper mainly considers a non-convex quadratic programming problem with convex quadratic constraints. Firstly, the objective function of the problem is reconstructed into a form composed of only one convex function and several linear functions by using the eigenvalue decomposition technique of matrix. Then the reconstructed problem is converted to the equivalent problem with simple concave quadratic objective function in the outcome space by introducing appropriate auxiliary variables, and its feasible domain is convex. Based on the branch-and-bound framework which can guarantee the global optimality of the solution, a global optimization algorithm for solving the equivalent problem is proposed, which integrates the effective relaxation process and the branching process related to the outer approximation technique. Finally, the effectiveness and feasibility of the algorithm are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

Sparse PCA is the optimization problem obtained from PCA by adding a sparsity constraint on the principal components. Sparse PCA is NP-hard and hard to approximate even in the single-component case. In this paper we settle the computational complexity of sparse PCA with respect to the rank of the covariance matrix. We show that, if the rank of the covariance matrix is a fixed value, then there is an algorithm that solves sparse PCA to global optimality, whose running time is polynomial in the number of features. We also prove a similar result for the version of sparse PCA which requires the principal components to have disjoint supports. [ABSTRACT FROM AUTHOR]

The total least squares problem with the general Tikhonov regularization can be reformulated as a one-dimensional parametric minimization problem (PM), where each parameterized function evaluation corresponds to solving an n-dimensional trust region subproblem. Under a mild assumption, the parametric function is differentiable and then an efficient bisection method has been proposed for solving (PM) in literature. In the first part of this paper, we show that the bisection algorithm can be greatly improved by reducing the initially estimated interval covering the optimal parameter. It is observed that the bisection method cannot guarantee to find the globally optimal solution since the nonconvex (PM) could have a local non-global minimizer. The main contribution of this paper is to propose an efficient branch-and-bound algorithm for globally solving (PM), based on a new underestimation of the parametric function over any given interval using only the information of the parametric function evaluations at the two endpoints. We can show that the new algorithm (BTD Algorithm) returns a global ϵ-approximation solution in a computational effort of at most O(n3/ϵ) under the same assumption as in the bisection method. The numerical results demonstrate that our new global optimization algorithm performs even much faster than the improved version of the bisection heuristic algorithm. [ABSTRACT FROM AUTHOR]

It is known [Mangasarian, A Newton method for linear programming, J. Optim. Theory Appl. 121 (2004), pp. 1–18] that every linear program can be solved exactly by minimizing an unconstrained quadratic penalty program. The penalty program is parameterized by a scalar t>0, and one is able to solve the original linear program in this manner when t is selected larger than a finite, but unknown t 0 > 0. In this paper, we show that every linear program can be solved using the solution to a parameter-free penalty program. We also characterize the solutions to the quadratic penalty programs using fixed points of certain nonexpansive maps. This leads to an iterative thresholding algorithm that converges to a desired limit point. We show in numerical experiments that this iterative method can outperform a variety of standard quadratic program solvers. Finally, we show that for every t ∈ R , the solution one obtains by solving a parameterized penalty program is guaranteed to lie in the feasible set of the original linear program. [ABSTRACT FROM AUTHOR]