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Properties of Structured Tensors and Complementarity Problems J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200212
Wei Mei, Qingzhi YangAbstract In this paper, we present some new results on a class of tensors, which are defined by the solvability of the corresponding tensor complementarity problem. For such structured tensors, we give a sufficient condition to guarantee the nonzero solution of the corresponding tensor complementarity problem with a vector containing at least two nonzero components and discuss their relationships with

A Modified Nonlinear Conjugate Gradient Algorithm for LargeScale Nonsmooth Convex Optimization J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200212
Tsegay Giday Woldu, Haibin Zhang, Xin Zhang, Yemane Hailu FissuhAbstract Nonlinear conjugate gradient methods are among the most preferable and effortless methods to solve smooth optimization problems. Due to their clarity and low memory requirements, they are more desirable for solving largescale smooth problems. Conjugate gradient methods make use of gradient and the previous direction information to determine the next search direction, and they require no numerical

Inexact Variable Metric Stochastic BlockCoordinate Descent for Regularized Optimization J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200229
Chingpei Lee, Stephen J. WrightAbstract Blockcoordinate descent is a popular framework for largescale regularized optimization problems with blockseparable structure. Existing methods have several limitations. They often assume that subproblems can be solved exactly at each iteration, which in practical terms usually restricts the quadratic term in the subproblem to be diagonal, thus losing most of the benefits of higherorder

Gradient Formulae for Nonlinear Probabilistic Constraints with Nonconvex Quadratic Forms J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200212
Wim van Ackooij, Pedro PérezArosAbstract Probability functions appearing in chance constraints are an ingredient of many practical applications. Understanding differentiability, and providing explicit formulae for gradients, allow us to build nonlinear programming methods for solving these optimization problems from practice. Unfortunately, differentiability of probability functions cannot be taken for granted. In this paper, motivated

On the Quality of FirstOrder Approximation of Functions with Hölder Continuous Gradient J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200211
Guillaume O. Berger, P.A. Absil, Raphaël M. Jungers, Yurii NesterovAbstract We show that Hölder continuity of the gradient is not only a sufficient condition, but also a necessary condition for the existence of a global upper bound on the error of the firstorder Taylor approximation. We also relate this global upper bound to the Hölder constant of the gradient. This relation is expressed as an interval, depending on the Hölder constant, in which the error of the

Optimality Conditions for Nonconvex Nonsmooth Optimization via Global Derivatives J. Optim. Theory Appl. (IF 1.6) Pub Date : 20191207
Felipe LaraAbstract The notions of upper and lower global directional derivatives are introduced for dealing with nonconvex and nonsmooth optimization problems. We provide calculus rules and monotonicity properties for these notions. As a consequence, new formulas for the Dini directional derivatives, radial epiderivatives and generalized asymptotic functions are given in terms of the upper and lower global directional

Explicit Formula for Preimages of Relaxed OneSided Lipschitz Mappings with Negative Lipschitz Constants: A Geometric Approach J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200228
Andrew S. Eberhard, Boris S. Mordukhovich, Janosch RiegerAbstract This paper addresses Lipschitzian stability issues, that play an important role in both theoretical and numerical aspects of variational analysis, optimization, and their applications. We particularly concentrate on the socalled relaxed onesided Lipschitz property of setvalued mappings with negative Lipschitz constants. This property has been much less investigated than more conventional

A General Iterative Procedure to Solve Generalized Equations with Differentiable Multifunction J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200211
Michaël Gaydu, Gilson N. SilvaAbstract Taking advantage of recent developments in the theory of generalized differentiation of multifunctions, we present in a unified manner a general iterative procedure for solving generalized equations. This procedure is based on a certain type of approximation of functions called pointbased approximation together with a linearization of the multifunctions. Our theorem encompasses the Newton

Poisson Stability and Periodicity of Control Affine Systems J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200229
Josiney A. SouzaAbstract This article discusses periodic and Poisson stable points of control affine systems. The main result states that the Poisson stability and the periodicity are equivalent concepts for states with closed semiorbits. This is applied to providing a sufficient condition for the existence of periodic trajectories. Almost periodicity of invariant control systems on Lie groups is specially studied

Connectedness of Solution Sets for Weak Generalized Symmetric Ky Fan Inequality Problems via AdditionInvariant Sets J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200217
Zaiyun Peng, Ziyuan Wang, Xinmin YangAbstract In this paper, the connectedness and pathconnectedness of solution sets for weak generalized symmetric Ky Fan inequality problems with respect to additioninvariant set are studied. A class of weak generalized symmetric Ky Fan inequality problems via additioninvariant set is proposed. By using a nonconvex separation theorem, the equivalence between the solutions set for the symmetric Ky

Optimality Conditions for DiscreteTime Control Problems J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200229
Marko Antonio RojasMedar, Camila Isoton, Lucelina Batista dos Santos, Violeta VivancoOrellanaAbstract We consider an optimal control problem governed by a system of nonlinear difference equations. We obtain the existence of the optimal control as well as firstorder optimality conditions of Pontryagin type by using the Dubovitskii–Milyutin formalism. Also, we give the necessary and sufficient conditions for global optimality.

Some Properties for Bisemivalues on Bicooperative Games J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200229
Margarita Domènech, José Miguel Giménez, María Albina PuenteAbstract In this work, we focus on bicooperative games, a variation of the classic cooperative games, and investigate the conditions for the coefficients of the bisemivalues—a generalization of semivalues for cooperative games—necessary and / or sufficient in order to satisfy some properties, including among others, desirability relation, balanced contributions, null player exclusion property and block

Analysis of a New Sequential Optimality Condition Applied to Mathematical Programs with Equilibrium Constraints J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200330
Elias S. Helou, Sandra A. Santos, Lucas E. A. SimõesAbstract In this study, a novel sequential optimality condition for general continuous optimization problems is established. In the context of mathematical programs with equilibrium constraints, the condition is proved to ensure Clarke stationarity. Originally devised for constrained nonsmooth optimization, the proposed sequential optimality condition addresses the domain of the constraints instead

Newton Method for Finding a Singularity of a Special Class of Locally Lipschitz Continuous Vector Fields on Riemannian Manifolds J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200330
Fabiana R. de Oliveira, Orizon P. FerreiraAbstract We extend some results of nonsmooth analysis from the Euclidean context to the Riemannian setting. Particularly, we discuss the concepts and some properties, such as the Clarke generalized covariant derivative, upper semicontinuity, and Rademacher theorem, of locally Lipschitz continuous vector fields on Riemannian settings. In addition, we present a version of the Newton method for finding

Optimal Control of HistoryDependent Evolution Inclusions with Applications to Frictional Contact J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200330
Stanisław MigórskiAbstract In this paper, we study a class of subdifferential evolution inclusions involving historydependent operators. First, we improve an existence and uniqueness theorem and prove the continuous dependence result in the weak topologies. Next, we establish the existence of optimal solution to an optimal control problem for the evolution inclusion. Finally, we illustrate the results by an example

Convergence of Solutions to Set Optimization Problems with the Set Less Order Relation J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200330
Lam Quoc Anh, Tran Quoc Duy, Dinh Vinh Hien, Daishi Kuroiwa, Narin PetrotAbstract This article investigates stability conditions for set optimization problems with the set less order relation in the senses of Panilevé–Kuratowski and Hausdorff convergence. Properties of various kinds of convergences for elements in the image space are discussed. Taking such properties into account, formulations of internal and external stability of the solutions are studied in the image

Regularization in Banach Spaces with Respect to the Bregman Distance J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200328
Mohamed Soueycatt, Yara Mohammad, Yamar HamwiAbstract The Moreau envelope, also known as Moreau–Yosida regularization, and the associated proximal mapping have been widely used in Hilbert and Banach spaces. They have been objects of great interest for optimizers since their conception more than half a century ago. They were generalized by the notion of the DMoreau envelope and Dproximal mapping by replacing the usual square of the Euclidean

New HigherOrder Strong Karush–Kuhn–Tucker Conditions for Proper Solutions in Nonsmooth Optimization J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200328
Nguyen Minh TungAbstract This paper considers higherorder necessary conditions for Henigproper, positively proper and Bensonproper solutions. Under suitable constraint qualifications, our conditions are of the Karush–Kuhn–Tucker rule. The conditions include higherorder complementarity slackness for both the objective and the constraining maps. They are in a nonclassical form with a supremum expression on the righthand

Smoothness Parameter of Power of Euclidean Norm J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200327
Anton Rodomanov, Yurii NesterovAbstract In this paper, we study derivatives of powers of Euclidean norm. We prove their Hölder continuity and establish explicit expressions for the corresponding constants. We show that these constants are optimal for odd derivatives and at most two times suboptimal for the even ones. In the particular case of integer powers, when the Hölder continuity transforms into the Lipschitz continuity, we

Analysis and Damage Identification of a Moderately Thick Cracked Beam Using an Interdependent LockingFree Element J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200326
Marco Pingaro, Giacomo Maurelli, Paolo VeniniAbstract The Timoshenko interdependent interpolation element, based on the assumption of cubic interpolation for the transverse displacement and quadratic interpolation for the rotation, is developed for both the static and the dynamic problems. Next, the different behavior of a beam due to the presence of a damaged zone is investigated and the problem of identifying diffused crack affecting a portion

Bregman Proximal Mappings and Bregman–Moreau Envelopes Under Relative ProxRegularity J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200131
Emanuel Laude, Peter Ochs, Daniel CremersAbstract We systematically study the local singlevaluedness of the Bregman proximal mapping and local smoothness of the Bregman–Moreau envelope of a nonconvex function under relative proxregularity—an extension of proxregularity—which was originally introduced by Poliquin and Rockafellar. As Bregman distances are asymmetric in general, in accordance with Bauschke et al., it is natural to consider

Improving the Convergence of Distributed Gradient Descent via Inexact Average Consensus J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200317
Bin Du, Jiazhen Zhou, Dengfeng SunAbstract It is observed that the inexact convergence of the wellknown distributed gradient descent algorithm can be caused by inaccuracy of consensus procedures. Motivated by this, to achieve the improved convergence, we ensure the sufficiently accurate consensus via approximate consensus steps. The accuracy is controlled by a predefined sequence of consensus error bounds. It is shown that one can

On the Numerical Solution of Differential Linear Matrix Inequalities J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200317
Marco Ariola, Gianmaria De Tommasi, Adriano Mele, Gaetano TartaglioneAbstract This paper presents a novel approach for the numerical solution of differential linear matrix inequalities. The solutions are searched in the class of piecewisequadratic functions with symmetric matrix coefficients to be determined. To limit the numbers of unknowns, congruence constraints are considered to guarantee continuity of the solution and of its derivative. In Example section, some

Representation of Hamilton–Jacobi Equation in Optimal Control Theory with Unbounded Control Set J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200317
Arkadiusz MisztelaAbstract In this paper, we study the existence of sufficiently regular representations of Hamilton–Jacobi equations in the optimal control theory with unbounded control set. We use a new method to construct representations for a wide class of Hamiltonians. This class is wider than any constructed before, because we do not require Legendre–Fenchel conjugates of Hamiltonians to be bounded. However, in

Necessary SecondOrder Conditions for a Strong Local Minimum in a Problem with Endpoint and Control Constraints J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200317
Nikolai Pavlovich OsmolovskiiAbstract The method of sliding modes (relaxation) was originally invented in optimal control in order to give a transparent proof of the maximum principle (a firstorder necessary condition for a strong local minimum) using the local maximum principle (a firstorder necessary condition for a weak local minimum). In the present work, we use this method to derive secondorder necessary conditions for

Suboptimal Control for Nonlinear Systems with Disturbance via Integral Sliding Mode Control and Policy Iteration J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200313
Yuqing Zheng, Guoshan ZhangAbstract In this paper, we consider a suboptimal control problem for nonlinear systems with unmatched disturbance via integral sliding mode control and policy iteration. Firstly, the unmatched disturbance is estimated by a nonlinear disturbance observer. Secondly, the integral sliding mode controller based on the disturbance estimation is designed to guarantee the reachability of the slidingmode surface

On the Split Equality Fixed Point Problem of QuasiPseudoContractive Mappings Without A Priori Knowledge of Operator Norms with Applications J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200312
Shihsen Chang, JenChih Yao, ChingFeng Wen, Liangcai ZhaoAbstract In this paper, we consider the split equality fixed point problem for quasipseudocontractive mappings without a priori knowledge of operator norms in Hilbert spaces, which includes split feasibility problem, split equality problem, split fixed point problem, etc., as special cases. A unified framework for the study of this kind of problems and operators is provided. The results presented

Semidefinite Program Duals for Separable Polynomial Programs Involving Box Constraints J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200304
Thai Doan ChuongAbstract We show that a separable polynomial program involving a box constraint enjoys a dual problem, that can be displayed in terms of sums of squares univariate polynomials. Under convexification and qualification conditions, we prove that a strong duality relation between the underlying separable polynomial problem and its corresponding dual holds, where the dual problem can be reformulated and

Normality and Nondegeneracy of the Maximum Principle in Optimal Impulsive Control Under State Constraints J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200304
Monica Motta, Caterina SartoriAbstract We investigate nondegenerate and normal forms of the maximum principle for general, free endtime, impulsive optimal control problems with state and endpoint constraints. We introduce constraint qualifications sufficient to avoid degeneracy or abnormality phenomena, which do not require any convexity and impose the existence of an inward pointing velocity just on the subset of times, in which

Nonemptiness and Compactness of Solution Sets to Generalized Polynomial Complementarity Problems J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200304
MengMeng Zheng, ZhengHai Huang, XiaoXiao MaAbstract In this paper, we investigate the generalized polynomial complementarity problem, which is a subclass of generalized complementarity problems with the involved map pairs being two polynomials. Based on the analysis on two structured tensor pairs located in the heading items of polynomials involved, and by using the degree theory, we achieve several results on the nonemptiness and compactness

A Note On the Weak Convergence of the Extragradient Method for Solving PseudoMonotone Variational Inequalities J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200302
Phan Tu VuongAbstract A corrigendum in Vuong (J Optim Theory Appl 176:399–409, 2018) is given.

Safe Feature Elimination for Nonnegativity Constrained Convex Optimization J. Optim. Theory Appl. (IF 1.6) Pub Date : 20191209
James Folberth, Stephen BeckerAbstract Inspired by recent work on safe feature elimination for 1norm regularized leastsquares, we develop strategies to eliminate features from convex optimization problems with nonnegativity constraints. Our strategy is safe in the sense that it will only remove features/coordinates from the problem when they are guaranteed to be zero at a solution. To perform feature elimination, we use an accurate

Sturm–Liouville Problems Involving Distribution Weights and an Application to Optimal Problems J. Optim. Theory Appl. (IF 1.6) Pub Date : 20190927
Hongjie Guo, Jiangang QiAbstract This paper is concerned with Sturm–Liouville problems (SLPs) with distribution weights and sets up the min–max principle and Lyapunovtype inequality for such problems. As an application, the paper solves the following optimization problems: If the first eigenvalue of a string vibration problem is known, what is the minimal total mass and by which distribution of weight is it attained; if

On Set Containment Characterizations for Sets Described by SetValued Maps with Applications J. Optim. Theory Appl. (IF 1.6) Pub Date : 20191108
Nithirat Sisarat, Rabian Wangkeeree, Gue Myung LeeAbstract In this paper, dual characterizations of the containment of two sets involving convex setvalued maps are investigated. These results are expressed in terms of the epigraph of a conjugate function of infima associated with corresponding setvalued maps. As an application, we establish characterizations of weak and proper efficient solutions of setvalued optimization problems in the sense

Combining Stochastic Adaptive Cubic Regularization with Negative Curvature for Nonconvex Optimization J. Optim. Theory Appl. (IF 1.6) Pub Date : 20191224
Seonho Park, Seung Hyun Jung, Panos M. PardalosAbstract We focus on minimizing nonconvex finitesum functions that typically arise in machine learning problems. In an attempt to solve this problem, the adaptive cubicregularized Newton method has shown its strong global convergence guarantees and the ability to escape from strict saddle points. In this paper, we expand this algorithm to incorporating the negative curvature method to update even

Linearized Methods for Tensor Complementarity Problems J. Optim. Theory Appl. (IF 1.6) Pub Date : 20200107
HongBo Guan, DongHui LiAbstract In this paper, we first propose a linearized method for solving the tensor complementarity problem. The subproblems of the method can be solved by solving linear complementarity problems with a constant matrix. We show that if the initial point is appropriately chosen, then the generated sequence of iterates converges to a solution of the problem monotonically. We then propose a lowerdimensional

On Minimal Copulas under the Concordance Order J. Optim. Theory Appl. (IF 1.6) Pub Date : 20191210
Jae Youn Ahn, Sebastian FuchsAbstract In the present paper, we study extreme negative dependence focussing on the concordance order for copulas. With the absence of a least element for dimensions \(d\ge 3\), the set of all minimal elements in the collection of all copulas turns out to be a natural and quite important extreme negative dependence concept. We investigate several sufficient conditions, and we provide a necessary condition

Finding the ForwardDouglas–RachfordForward Method J. Optim. Theory Appl. (IF 1.6) Pub Date : 20191031
Ernest K. Ryu, Bằng Công VũAbstract We consider the monotone inclusion problem with a sum of 3 operators, in which 2 are monotone and 1 is monotoneLipschitz. The classical Douglas–Rachford and forward–backward–forward methods, respectively, solve the monotone inclusion problem with a sum of 2 monotone operators and a sum of 1 monotone and 1 monotoneLipschitz operators. We first present a method that naturally combines Douglas–Rachford

Matrix Optimization Over LowRank Spectral Sets: Stationary Points and Local and Global Minimizers J. Optim. Theory Appl. (IF 1.6) Pub Date : 20191209
Xinrong Li, Naihua Xiu, Shenglong ZhouAbstract In this paper, we consider matrix optimization with the variable as a matrix that is constrained into a lowrank spectral set, where the lowrank spectral set is the intersection of a lowrank set and a spectral set. Three typical spectral sets are considered, yielding three lowrank spectral sets. For each lowrank spectral set, we first calculate the projection of a given point onto this

Robust Suboptimality of LinearSaturated Control via Quadratic ZeroSum Differential Games J. Optim. Theory Appl. (IF 1.6) Pub Date : 20191209
Dario Bauso, Rosario Maggistro, Raffaele PesentiAbstract In this paper, we determine the approximation ratio of a linearsaturated control policy of a typical robuststabilization problem. We consider a system, whose state integrates the discrepancy between the unknown but bounded disturbance and control. The control aims at keeping the state within a target set, whereas the disturbance aims at pushing the state outside of the target set by opposing

Inertial ProjectionType Methods for Solving QuasiVariational Inequalities in Real Hilbert Spaces J. Optim. Theory Appl. (IF 1.6) Pub Date : 20191207
Yekini Shehu, Aviv Gibali, Simone SagratellaAbstract In this paper, we introduce an inertial projectiontype method with different updating strategies for solving quasivariational inequalities with strongly monotone and Lipschitz continuous operators in real Hilbert spaces. Under standard assumptions, we establish different strong convergence results for the proposed algorithm. Primary numerical experiments demonstrate the potential applicability

Using Age Structure for a Multistage Optimal Control Model with Random Switching Time. J. Optim. Theory Appl. (IF 1.6) Pub Date : 20191207
Stefan Wrzaczek,Michael Kuhn,Ivan FrankovicThe paper presents a transformation of a multistage optimal control model with random switching time to an agestructured optimal control model. Following the mathematical transformation, the advantages of the present approach, as compared to a standard backward approach, are discussed. They relate in particular to a compact and unified representation of the two stages of the model: the applicability

Stochastic Mirror Descent Dynamics and Their Convergence in Monotone Variational Inequalities. J. Optim. Theory Appl. (IF 1.6) Pub Date : 20181113
Panayotis Mertikopoulos,Mathias StaudiglWe examine a class of stochastic mirror descent dynamics in the context of monotone variational inequalities (including Nash equilibrium and saddlepoint problems). The dynamics under study are formulated as a stochastic differential equation, driven by a (singlevalued) monotone operator and perturbed by a Brownian motion. The system's controllable parameters are two variable weight sequences, that

On the Convergence Analysis of the Optimized Gradient Method. J. Optim. Theory Appl. (IF 1.6) Pub Date : 20170504
Donghwan Kim,Jeffrey A FesslerThis paper considers the problem of unconstrained minimization of smooth convex functions having Lipschitz continuous gradients with known Lipschitz constant. We recently proposed the optimized gradient method for this problem and showed that it has a worstcase convergence bound for the cost function decrease that is twice as small as that of Nesterov's fast gradient method, yet has a similarly efficient

Stationary Anonymous Sequential Games with Undiscounted Rewards. J. Optim. Theory Appl. (IF 1.6) Pub Date : 20150825
Piotr Więcek,Eitan AltmanStationary anonymous sequential games with undiscounted rewards are a special class of games that combine features from both population games (infinitely many players) with stochastic games. We extend the theory for these games to the cases of total expected reward as well as to the expected average reward. We show that in the anonymous sequential game equilibria correspond to the limits of those of

The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space. J. Optim. Theory Appl. (IF 1.6) Pub Date : 20110415
Y Censor,A Gibali,S ReichWe present a subgradient extragradient method for solving variational inequalities in Hilbert space. In addition, we propose a modified version of our algorithm that finds a solution of a variational inequality which is also a fixed point of a given nonexpansive mapping. We establish weak convergence theorems for both algorithms.

Isotonic Regression under Lipschitz Constraint. J. Optim. Theory Appl. (IF 1.6) Pub Date : 20090501
L Yeganova,W J WilburThe pool adjacent violators (PAV) algorithm is an efficient technique for the class of isotonic regression problems with complete ordering. The algorithm yields a stepwise isotonic estimate which approximates the function and assigns maximum likelihood to the data. However, if one has reasons to believe that the data were generated by a continuous function, a smoother estimate may provide a better

Convergence Rates of ForwardDouglasRachford Splitting Method. J. Optim. Theory Appl. (IF 1.6) Pub Date : null
Cesare Molinari,Jingwei Liang,Jalal FadiliOver the past decades, operator splitting methods have become ubiquitous for nonsmooth optimization owing to their simplicity and efficiency. In this paper, we consider the ForwardDouglasRachford splitting method and study both global and local convergence rates of this method. For the global rate, we establish a sublinear convergence rate in terms of a Bregman divergence suitably designed for the

The Proximal Alternating Minimization Algorithm for TwoBlock Separable Convex Optimization Problems with Linear Constraints. J. Optim. Theory Appl. (IF 1.6) Pub Date : null
Sandy Bitterlich,Radu Ioan Boţ,Ernö Robert Csetnek,Gert WankaThe Alternating Minimization Algorithm has been proposed by Paul Tseng to solve convex programming problems with twoblock separable linear constraints and objectives, whereby (at least) one of the components of the latter is assumed to be strongly convex. The fact that one of the subproblems to be solved within the iteration process of this method does not usually correspond to the calculation of