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In this paper, we introduce a modified Tseng's extragradient method with a new step-length rule to solve pseudo-monotone variational inequalities in real Hilbert spaces. Under suitable conditions, the sequence generated by this algorithm strongly converges to the common elements of the solution set of pseudo-monotone variational inequality problems and the fixed point set of k-demicontractive mappings. Finally, we give some numerical experiments to illustrate the effectiveness of the proposed algorithm. The main results of this paper generalize and improve some known results in the literature. [ABSTRACT FROM AUTHOR]

The growing use of optimization models to help decision making has created a demand for such tools that allow formulating and solving more models of real-world processes and systems related to human activity in which hypotheses are not verify in a way specific for classical optimization. One of the approaches for real-world extremum problems under uncertainty is interval-valued optimization. In this paper, a twice differentiable vector optimization problem with multiple interval-valued objective function and both inequality and equality constraints is considered. In this paper, the first order necessary optimality conditions of Karush-Kuhn-Tucker type are proved for differentiable interval-valued vector optimization problems under the first order constraint qualification. If the interval-valued objective function is assumed to be twice weakly differentiable and constraints functions are assumed to be twice differentiable, then two types of second order necessary optimality conditions under two various constraint qualifications are proved for such smooth interval-valued vector optimization problems. Finally, in order to illustrate the Karush-Kuhn-Tucker type necessary optimality conditions established in the paper, an example of an interval-valued optimization is given. [ABSTRACT FROM AUTHOR]

The problem considered in this paper is a sequence of two optimization problems in which the feasible region of the upper-level optimization problem is determined implicitly by the solution set of the lower-level optimization problem. Using the optimal value reformulation, together with the partial calmness property, we give necessary optimality conditions in terms of tangential subdifferentials. An example that illustrates our finding is also given. [ABSTRACT FROM AUTHOR]

In this paper, a numerical approximation method is developed to find approximate solutions to a class of constrained multi-objective optimization problems. All the functions of the problem are not necessarily convex functions. At each iteration of the method, a particular type of subproblem is solved using the trust region technique, and the step is evaluated using the notions of actual reduction and predicted reduction. A non-differentiable $ l_{\infty } $ l∞ penalty function restricts the constraint violations. An adaptive BFGS update formula is introduced. Global convergence of the proposed algorithm is established under the Mangasarian-Fromovitz constraint qualification and some mild assumptions. Furthermore, it is justified that the proposed algorithm displays a super-linear convergence rate. Numerical results are provided to show the efficiency of the algorithm in the quality of the approximated Pareto front. [ABSTRACT FROM AUTHOR]

AbstractConjugate gradient methods are probably the most used methods in solving large scale unconstrained optimization problems. They have become popular because of their simplicity and low memory requirements. In this paper, we propose a hybrid conjugate gradient method based on the improved Fletcher-Reeves (IFR) and Rivaie-Mustafa-Ismail-Leong+ (RMIL+) methods and establish its global convergence under the strong Wolfe line search conditions. The new conjugate gradient direction satisfies the sufficient descent condition. The method is then compared to other methods in the literature and numerical experiments show that it is competent when solving large scale unconstrained optimization problems. Furthermore, the method is applied to solve a problem in portfolio selection. [ABSTRACT FROM AUTHOR]

In this article, two Hager–Zhang (HZ) type projection algorithms are presented for large-dimension nonlinear monotone problems and sparse signal recovery in compressed sensing. This goal is attained by conducting singular value analysis of a nonsingular HZ-type search direction matrix as well as applying the idea by Piazza and Politi [J Comput Appl Math. 2002;143(1):141–144] and minimizing the Frobenius norm of an orthornormal matrix. The paper attempts to fill the gap in the work of Hager and Zhang [Pac J Optim. 2006;2(1):35–58], Waziri et al. [Appl Math Comput. 2019;361:645–660], Sabi'u et al. [Appl Numer Math. 2020;153:217–233] and Babaie-Kafaki [4OR-Q J Oper Res. 2014;12:285-292], where the sufficient descent or global convergence condition is not satisfied when the HZ parameter is in the interval $ (0,\frac {1}{4}) $ (0 , 1 4). The proposed schemes are also suitable for solving non-smooth nonlinear problems. Also, by employing some mild conditions, global convergence of the schemes are established, while numerical comparison with four effective HZ-type methods show that the new methods are efficient. Furthermore, to illustrate their practical application, both methods are applied to solve the $ \ell _1 $ ℓ 1 -norm regularization problems to recover a sparse signal in compressive sensing. The experiments conducted in that regard show that the methods are promising and perform better than two other methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper, a general formula concerning the multi-composition rule of convex subdifferential calculus is provided in the setting of Banach spaces under an appropriate regularity condition. As an application, this calculus rule is applied to obtain necessary and sufficient Karush–Kuhn–Tucker type optimality conditions for constrained convex minmax location problems with perturbed minimal time functions and set-up costs. [ABSTRACT FROM AUTHOR]

In this two-part study, we develop a general theory of the so-called exact augmented Lagrangians for constrained optimization problems in Hilbert spaces. In contrast to traditional nonsmooth exact penalty functions, these augmented Lagrangians are continuously differentiable for smooth problems and do not suffer from the Maratos effect, which makes them especially appealing for applications in numerical optimization. Our aim is to present a detailed study of various theoretical properties of exact augmented Lagrangians and discuss several applications of these functions to constrained variational problems, problems with PDE constraints, and optimal control problems. The first paper is devoted to a theoretical analysis of an exact augmented Lagrangian for optimization problems in Hilbert spaces. We obtain several useful estimates of this augmented Lagrangian and its gradient, and present several types of sufficient conditions for KKT-points of a constrained problem corresponding to locally/globally optimal solutions to be local/global minimizers of the exact augmented Lagrangian. [ABSTRACT FROM AUTHOR]

This paper demonstrates the optimality of an interpolation set employed in derivative-free trust-region methods. This set is optimal in the sense that it minimizes the constant of well-poisedness in a ball centred at the starting point. It is chosen as the default initial interpolation set by many derivative-free trust-region methods based on underdetermined quadratic interpolation, including NEWUOA, BOBYQA, LINCOA, and COBYQA. Our analysis provides a theoretical justification for this choice. [ABSTRACT FROM AUTHOR]

In this paper, we propose a reflected forward-backward splitting algorithm with two different inertial extrapolations to find a zero of the sum of three monotone operators consisting of the maximal monotone operator, Lipschitz continuous monotone operator, and a cocoercive operator in real Hilbert spaces. One of the interesting features of the proposed algorithm is that both the Lipschitz continuous monotone operator and the cocoercive operator are computed explicitly each with one evaluation per iteration. We then obtain weak and strong convergence results under some mild conditions. We finally give a numerical illustration to show that our proposed algorithm is effective and competitive with other related algorithms in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlinear scalarization function for a variable domination structure in an arbitrary linear space without assuming any particular topology. Conditions are provided under which the nonlinear scalarization function possesses several useful properties such as finiteness, properness, positive homogeneity, subadditivity, (strict) monotonicity, convexity or continuity. These properties are employed to characterize approximate efficiency in linear spaces. [ABSTRACT FROM AUTHOR]

The main aim of the paper is to introduce certain constraint qualifications for a nonsmooth programming problem in terms of semi-regular convexificators and investigate their relations with other existing notions of constraint qualifications. The programming problem under consideration has mixed constraints, that is, it involves both inequality and equality constraints. All these notions are in terms of upper semi-regular convexificators of inequality constraints and pseudo-differentials of equality constraints. Based on a sufficient condition for error bound property, the implication relation between quasinormality and error bound property in terms of convexificators is investigated in this paper. Three conditions are introduced, namely constant positive linear dependence condition (CPLD), constant rank constraint qualification (CRCQ) and Mangasarian–Fromovitz constraint qualification (MFCQ) in terms of convexificators. These conditions are in fact shown to be constraint qualifications as Karush–Kuhn–Tucker optimality conditions hold when CPLD holds and both MFCQ and CRCQ imply CPLD. Further, it is observed that CPLD and quasinormality conditions are independent for nonsmooth problems in terms of convexificators. [ABSTRACT FROM AUTHOR]

In this paper, we establish a minimal criterion of efficiency for a class of multiobjective variational control problems. Thus, some minimal conditions are formulated so that all local efficient solutions are also global. Also, we present an example to illustrate the mathematical developments derived in the paper. [ABSTRACT FROM AUTHOR]

The traditional BFGS algorithm has been proved very efficient. It is convergent for convex nonlinear optimization problems. However, for non-convex nonlinear optimization problems, it is known that the BFGS algorithm may not be convergent. This paper proposes a robust BFGS algorithm in the sense that the algorithm superlinearly converges to a local minimum under some mild assumptions for both convex and non-convex nonlinear optimization problems. Numerical test on the CUTEst test set is reported to demonstrate the merit of the proposed robust BFGS algorithm. This result shows that the robust BFGS algorithm is very efficient and effective. [ABSTRACT FROM AUTHOR]

In this paper, a random derivative-free modified tensor method with curvilinear linesearch technique is considered for solving nonlinear programming problems. The proposed algorithm is designed to build polynomial interpolation models for the objective function and build the tensor model using the information of the interpolation function. At the same time, we give a new curvilinear tensor step which guarantees the monotonic decrease on the tensor model. The modified tensor step also asymptotically approaches the modified Newton direction as the step length shrinks to zero, and the objective function of problem will be descendent. Under general assumptions, we give the global and local superlinear convergence of the algorithm. Numerical results are recorded, and the compare results with a tensor algorithm without curvilinear linesearch technique and Newton algorithm show that our algorithm is more effectiveness. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the globally proper efficiency of set optimization problems. Firstly, we use the so-called certainly set less order relation to define a new kind of set order relation. Based on the new set order relation, we introduce the notion of the globally proper efficient solution of the set optimization problem. Secondly, we establish Lagrange multiplier rule of the set optimization problem. Finally, we obtain Lagrangian duality theorems and saddle point theorems. We also give some examples to illustrate our results. [ABSTRACT FROM AUTHOR]

The goal of this paper is to investigate an approach for derivative-free optimization that has not received sufficient attention in the literature and is yet one of the simplest to implement and parallelize. In its simplest form, it consists of employing derivative-based methods for unconstrained or constrained optimization and replacing the gradient of the objective (and constraints) by finite-difference approximations. This approach is applicable to problems with or without noise in the functions. The differencing interval is determined by a bound on the second (or third) derivative and by the noise level, which is assumed to be known or to be accessible through difference tables or sampling. The use of finite-difference gradient approximations has been largely dismissed in the derivative-free optimization literature as too expensive in terms of function evaluations or as impractical in the presence of noise. However, the test results presented in this paper suggest that it has much to be recommended. The experiments compare newuoa, dfo-ls and cobyla against finite-difference versions of l-bfgs, lmder and knitro on three classes of problems: general unconstrained problems, nonlinear least squares problems and nonlinear programs with inequality constraints. [ABSTRACT FROM AUTHOR]

In this paper, we propose a single timescale stochastic quasi-Newton method for solving the stochastic optimization problems. The objective function of the problem is a composition of two smooth functions and their derivatives are not available. The algorithm sets to approximate sequences to estimate the gradient of the composite objective function and the inner function. The matrix correction parameters are given in BFGS update form for avoiding the assumption that Hessian matrix of objective is positive definite. We show the global convergence of the algorithm. The algorithm achieves the complexity O (ϵ − 1) to find an ϵ − approximate stationary point and ensure that the expectation of the squared norm of the gradient is smaller than the given accuracy tolerance ϵ. The numerical results of nonconvex binary classification problem using the support vector machine and a multicall classification problem using neural networks are reported to show the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new nonlinear optimization model to solve semidefinite optimization problems (SDPs), providing some properties related to local optimal solutions. The proposed model is based on another nonlinear optimization model given by [S. Burer and R. Monteiro, A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization, Math. Program. Ser. B 95 (2003), pp. 329–357], but it has several nice properties not seen in the existing one. Firstly, the decision variable of the proposed model is a triangular low-rank matrix. Secondly, the existence of a strict local optimum of the proposed model is guaranteed under some conditions, whereas the existing model has no strict local optimum. In other words, it is difficult to construct solution methods equipped with fast convergence using the existing model. We also present some numerical results, showing that the use of the proposed model allows to deliver highly accurate solutions. [ABSTRACT FROM AUTHOR]

Optimization problems with least constraint violation have many important practical backgrounds. The related works in the literature normally focus on the norm square measure function and the classical augmented Lagrangian method is proved to be suitable for finding an approximate solution. This paper considers the constrained convex optimization problem with the least constraint violation under a general measure function. The properties of the conjugate dual associated with the measure function of the shifted problem are discussed through the relations between the dual function and the optimal value function. The differentiability of the dual function associated with the measure function is proved. The properties of augmented Lagrangian method induced by the measure function are characterized in terms of the dual function. The optimality conditions for the problem with the least constraint violation are established in term of the augmented Lagrangian. It is shown that the augmented Lagrangian method has the properties that the sequence of shift measure values is decreasing, the sequence of multipliers is unbounded, and the sequence of shifts converges to the least violated shift under an extra assumption. [ABSTRACT FROM AUTHOR]

Asymptotic stationarity and regularity conditions turned out to be quite useful to study the qualitative properties of numerical solution methods for standard nonlinear and complementarity constrained programs. In this paper, we first extend these notions to nonlinear optimization problems with nonsmooth but Lipschitzian data functions in order to find reasonable notions of asymptotic stationarity and regularity in terms of Clarke's and Mordukhovich's subdifferential construction. Particularly, we compare the associated novel asymptotic constraint qualifications with already existing ones. The second part of the paper presents two applications of the obtained theory. On the one hand, we specify our findings for complementarity constrained optimization problems and recover recent results from the literature which demonstrates the power of the approach. Furthermore, we hint at potential extensions to or- and vanishing constrained optimization. On the other hand, we demonstrate the usefulness of asymptotic regularity in the context of bilevel optimization. More precisely, we justify a well-known stationarity system for affinely constrained bilevel optimization problems in a novel way. Afterwards, we suggest a solution algorithm for this class of bilevel optimization problems which combines a penalty method with ideas from DC-programming. After a brief convergence analysis, we present the results of some numerical experiments. [ABSTRACT FROM AUTHOR]

The ℓ 1 -regularized minimization has been widely used in many data science applications, and certain special constrained ℓ 1 -regularized minimizations have also been proposed in some recent applications. In this paper, we consider a sphere constrained ℓ 1 -regularized minimization, which can arise in image processing, signal recognition and sparse principal component analysis. Viewing the sphere as a simple Riemannian manifold, manifold-based methods for non-smooth minimization can be applied to solve such a problem, but may still encounter slow convergence in some situations. Our objective of this paper is to propose a new and efficient active-set proximal quasi-Newton method for this problem. The idea behind is to speed up the convergence by separately handling the convergence of both the active and inactive variables. In particular, our method invokes a procedure to effectively estimate active and inactive variables, and then designs the search directions based on proximal gradients and quasi-Newton directions to efficiently treat the convergence of the active and inactive variables, respectively. We show that under some mild conditions, the global convergence is guaranteed, and the complexity is also performed to reveal the computational efficiency. Numerical experience on the ℓ 1 -regularized quadratic programming and sparse principal component analysis on both synthetic and real data demonstrates its robustness and efficiency. [ABSTRACT FROM AUTHOR]

Projected-search methods for bound-constrained optimization are based on performing a search along a piecewise-linear continuous path obtained by projecting a search direction onto the feasible region. A potential benefit of a projected-search method is that many changes to the active set can be made at the cost of computing a single search direction. As the objective function is not differentiable along the search path, it is not possible to use a projected-search method with a step that satisfies the Wolfe conditions, which require the directional derivative of the objective function at a point on the path. For this reason, methods based in full or in part on a simple backtracking procedure must be used to give a step that satisfies an ‘Armijo-like’ sufficient decrease condition. As a consequence, conventional projected-search methods are unable to exploit sophisticated safeguarded polynomial interpolation techniques that have been shown to be effective for the unconstrained case. This paper describes a new framework for the development of a general class of projected-search methods for bound-constrained optimization. At each iteration, a descent direction is computed with respect to a certain extended active set. This direction is used to specify a search direction that is used in conjunction with a step length computed by a quasi-Wolfe search. The quasi-Wolfe search is designed specifically for use with a piecewise-linear search path and is similar to a conventional Wolfe line search, except that a step is accepted under a wider range of conditions. These conditions take into consideration steps at which the restriction of the objective function on the search path is not differentiable. Standard existence and convergence results associated with a conventional Wolfe line search are extended to the quasi-Wolfe case. In addition, it is shown that under a standard nondegeneracy assumption, any method within the framework will identify the optimal active set in a finite number of iterations. Computational results are given for a specific projected-search method that uses a limited-memory quasi-Newton approximation of the Hessian. The results show that, in this context, a quasi-Wolfe search is substantially more efficient and reliable than an Armijo-like search based on simple backtracking. Comparisons with a state-of-the-art bound-constrained optimization package are also presented. [ABSTRACT FROM AUTHOR]

In this paper, a class of E-differentiable multiobjective programming with multiple objective functions and with both inequality and equality constraints is considered. The so-called E-Karush–Kuhn–Tucker necessary optimality conditions for a (weak LS-Pareto) LS-Pareto solution are established for the considered E-differentiable vector optimization problem with the multiple interval-valued objective functions. Also several sufficient optimality conditions for both LU and LS order relations are derived for such interval-valued vector optimization problems under (generalized) E-invexity hypotheses. [ABSTRACT FROM AUTHOR]

In this paper we establish some approximation versions of the classical Dinkelbach algorithm for nonlinear fractional optimization problems in Banach spaces. We start by examining what occurs if at any step of the algorithm, the generated point desired to be a minimizer can only be determined with a given error. Next we assume that the step error tends to zero as the algorithm advances. The last version of the algorithm we present is making use of Ekeland's variational principle for generating the sequence of minimizer-like points. In the final part of the article we deliver some results in order to achieve a Palais-Smale type compactness condition that guarantees the convergence of our Dinkelbach-Ekeland algorithm. [ABSTRACT FROM AUTHOR]

This paper addresses the worst-case evaluation complexity of a version of the standard quadratic penalty method for smooth nonconvex optimization problems with constraints. The method analysed allows inexact solution of the subproblems and do not require prior knowledge of the Lipschitz constants related with the problem. When an approximate feasible point is used as starting point, it is shown that the referred method takes at most O (log ⁡ (σ 0 − 1 ϵ − 2)) outer iterations to generate an ϵ-approximate KKT point, where σ 0 is the first penalty parameter. For equality constrained problems, this bound yields to an evaluation complexity bound of O (ϵ − 4) , when σ 0 = ϵ − 2 and suitable first-order methods are used as inner solvers. For problems having only linear equality constraints, an evaluation complexity bound of O (ϵ − (p + 1) / p) is established when appropriate p-order methods ( p ≥ 2) are used as inner solvers. Illustrative numerical results are also presented and corroborate the theoretical predictions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate robust necessary optimality for scalar uncertain optimization problems under the strictly robust counterpart, where the uncertainties are included in the objective and the constraints. Based on generalized differentiable assumptions, some robust necessary conditions via a separation scheme are derived. Besides, by means of strong separation functions, several characterizations of robust necessary optimality can be established. Finally, an example is given to show the effectiveness of the results derived in this paper. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study some new models of nonlinear regularity on fixed sets of set-valued mappings defined on the complete metric spaces. Slope and coderivative characterizations of these models are given. The stability of the Milyutin regularity is investigated when the initial set-valued mapping is perturbed by a suitable Lipschitz single-valued map. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of constrained optimization problems whose constraints involve a cardinality or rank constraint. The penalty formulation based on a partial regularization has recently been promoted in the literature to approximate these problems, which usually outperforms the penalty formulation based on a full regularization in terms of solution quality. Nevertheless, the relation between the penalty formulation with a partial regularizer and the original problem was not much studied yet. Under some suitable assumptions, we show that the penalty formulation based on a partial regularization is an exact reformulation of the original problem in the sense that they both share the same global minimizers. We also show that a local minimizer of the original problem is that of the penalty reformulation. These results provide some theoretical justification for the often-observed superior performance of the penalty model based on a partial regularizer over a corresponding full regularizer. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new nonlinear scalarization function and apply it to investigate optimality conditions and scalar representations for minimal sets and set optimization problems with the set less order relation. First, we propose a new scalarization function for sets belonging to the power set of a normed space and discuss its properties. Next, we apply such properties to formulate sufficient and necessary optimality conditions for strictly minimal and weakly minimal sets. Finally, under suitable adjustments, we provide nonlinear scalarization functions and investigate scalar representations of strictly efficient solution and weakly efficient solution sets of the reference problems. [ABSTRACT FROM AUTHOR]

Supply chain network design (SCND) is the process for designing and modeling the supply chain such that to minimize the costs generated by the location of facilities and the flow of product between the selected facilities. The aim of this paper is to investigate a particular SCND, namely the two-stage supply chain network design problem with risk-pooling and lead times and to provide a novel efficient and effective genetic algorithm (GA) designed to fit the challenges of the considered optimization problem. Extensive computational experiments were performed on two sets of instances and the achieved results prove the performance of our proposed GA. [ABSTRACT FROM AUTHOR]

In this paper, inertial splitting algorithms for nonlinear operators of pseudocontractive and accretive types are proposed. Weak and strong convergence theorems are established in uniformly convex and q-uniformly smooth Banach spaces. Numerical examples are given to illustrate the effectiveness of our proposed algorithms. [ABSTRACT FROM AUTHOR]

Many real-world problems in applied sciences, engineering and economics can be reformulated as the convex minimization problem of two objective functions. In order to solve this problem, the forward–backward splitting algorithm and its modifications have been invented and used to investigate the convergence analysis. In this work, we discuss strong convergence of the sequence generated by the forward–backward algorithm involving the viscosity approximation method. The stepsizes studied in this paper are defined by two different kinds of linesearches which do not require the Lipschitz continuity condition on the gradient. Finally, we provide numerical experiments to show the efficiency of the proposed methods in signal recovery. It is reported that our algorithms have a good convergence behavior in practice without information of Lipschitz constants of the gradient of functions. [ABSTRACT FROM AUTHOR]

In this paper, a new three-term spectral conjugate subgradient method is presented to solve an absolute value equation. The new method is constructed based on conjugate gradient method proposed by Dai & Yuan and Barzilai & Borwein technique. In each iteration, a descent direction is generated. The global convergence of the new method is established under some mild assumptions. The numerical results are reported, showing the efficiency of our method. [ABSTRACT FROM AUTHOR]

In this study, we are interested in the lower bounds of primal iterates generated in an inner loop of a class of stabilized interior point methods. By using eigenvalue estimates and the perturbation theory of linear systems, we show that for a fixed barrier parameter, the sequence of primal iterates is bounded away from zero. The results are established under some weak assumptions, which are implied by, for example, a second-order sufficient condition. Moreover, our analysis are independent of the choices of merit functions and penalty factors. Preliminary numerical experiments on some degenerate problems conform the results of this paper. [ABSTRACT FROM AUTHOR]

Switching-constrained optimization problems form a difficult class of mathematical programmes since their feasible set is almost disconnected while standard constraint qualifications are likely to fail at several feasible points. That is why the application of standard methods from nonlinear programming does not seem to be promising in order to solve such problems. In this paper, we adapt several relaxation methods which are well known from the numerical treatment of mathematical programmes with complementarity constraints to the setting of switching-constrained optimization. A detailed convergence analysis is provided for the adapted relaxation schemes of Scholtes as well as Kanzow and Schwartz. While Scholtes' method and the relaxation scheme of Steffensen and Ulbrich only find weakly stationary points in general, it is shown that the adapted relaxation scheme of Kanzow and Schwartz is capable of identifying Mordukhovich-stationary points of switching-constrained programmes under suitable assumptions. Some computational experiments and a numerical comparison of the proposed methods based on examples from logical programming, switching control, and portfolio optimization close the paper. [ABSTRACT FROM AUTHOR]

The paper illustrates connections between classical results of Analysis and Optimization. The focus is on new elementary proofs of Implicit Function Theorem, Lusternik's Theorem, and optimality conditions for equality constrained optimization problems. The proofs are based on Fermat's Theorem and the Weierstrass Theorem and do not use the contraction mapping principle or other advanced results of Real Analysis, so they can be used in any introductory course on Optimization or Real Analysis without the requirement of the advanced background in analysis. The paper also presents a simple proof of Implicit Function Theorem in normed linear spaces. [ABSTRACT FROM AUTHOR]

In this paper, we propose a feasible and effective iteration method for solving the tensor equation A ∗ N X ∗ M B = C with a tensor inequality constraint D ∗ N X ∗ M E ≥ F . The global convergence is established. Numerical examples are provided to illustrate the feasibility and efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, weak sharp solutions are investigated for a class of variational-type inequalities involving curvilinear integral functionals. Moreover, by using the minimum principle sufficiency property, some characterization results on the weak sharpness property of the solution set for the considered class of variational-type inequalities are formulated and proved. Finally, in order to illustrate the main results, an application is provided. [ABSTRACT FROM AUTHOR]

In this paper, vector continuous-time programming problems with inequality type of constraints are considered. All the results are obtained under convexity assumptions. The main tool employed in the derivation of the zero order optimality conditions is a new version of theorem of the alternative in functional spaces. To apply the alternative theorem, a certain regularity condition must be satisfied. No differentiability assumption is imposed. [ABSTRACT FROM AUTHOR]

In this paper, we propose new numerical algorithms in the setting of unconstrained optimization problems and we prove the discrete rate of convergence of order O 1 / n 2 in the iterates of the convex objective function. Our optimization algorithms are obtained via discretizations from dynamical systems with Hessian-driven damping. Finally, some numerical experiments are presented in order to validate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of mixed variational inequalities on nonempty closed convex subsets of real Euclidean spaces. One of the mappings involved is lower semicontinuous and the other is weakly homogeneous. After discussing the boundedness for the solution set (if it is nonempty) of the problem, we focus on the nonemptiness and compactness of the solution set. Two new results on the nonemptiness and compactness of the solution set of the problem are established, and some examples are used to compare the results with those in the literature. It can be seen that new results improve some known related results. [ABSTRACT FROM AUTHOR]

In this paper, a derivative-free spectral projection technique to solve a system of large-scale nonlinear monotone equations is presented. The primary motivation is to use the appropriate structure of spectral conjugate gradient directions in the projection algorithms. The new direction is derivative-free and requires a little storage and computation. So, it is an appropriate direction to use in large-scale projection algorithms. We prove the global convergence and R-linear convergence rate of the proposed algorithm under some suitable conditions. Numerical experiments show a promising behaviour of the proposed algorithm to deal with large-scale monotone equations. Additionally, as a practical application, we use the new method to solve the l 1 -norm regularization problems to reconstruct a sparse signal in compressed sensing. [ABSTRACT FROM AUTHOR]

In this paper, we study various nonsmooth constraint qualifications for a system of inequalities with tangentially convex functions. This is done by using the notion of tangential subdifferentials together with various nonsmooth (generalized) constraint qualifications that are a modification of the well known constraint qualifications. Indeed, we establish new links among various generalized constraint qualifications. We also give a table summarizing various links among the generalized constraint qualifications. Several illustrative examples are presented to clarify the connection between nonsmooth constraint qualifications. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel dual active-set algorithm that is based on proximal gradient and semi-smooth Newton iterations for the sparse approximation of correlation matrices in the Frobenius norm. A new dual formulation with upper and lower bounds is derived. To solve the dual, the proximal gradient method is developed to guarantee global convergence. Also, it provides information to estimate active/inactive constraints. Then, the semi-smooth Newton method is applied to accelerate the convergence of the proximal gradient method, which is the key ingredient of our algorithm. It is shown that the proposed algorithm for the dual is globally convergent under certain conditions. Some preliminary numerical results are given to illustrate the effectiveness of our algorithm on synthetic and real data sets. [ABSTRACT FROM AUTHOR]

The paper introduces two new numerical methods for solving a variational inequality problem whose constraint set is expressed as the solution set of a monotone and Lipschitz-type equilibrium problem in a Hilbert space. We present how to combine regularization terms in an extragradient method and prove that the iterative sequences generated by the resulting methods converge strongly to a solution of equilibrium problem which solves the associated variational inequality problem. Theorems of strong convergence are analysed which are based on the incorporated Tikhonov regularization method for equilibrium problems. The first method is designed in the case where the Lipschitz-type constants of bifunction are known. While the second method can be implemented more easily without the prior knownledge of Lipschitz-type constants. The reason is that the second method have used a new stepsize rule whose computation is simple and easy to check at each step. Several numerical experiments are performed and they have demonstrated the effectiveness and the fast convergence of the new methods over existing methods. [ABSTRACT FROM AUTHOR]

This paper proposes a novel quadratic convex reformulation (QCR) for the nonconvex quadratic program with convex quadratic constraints. This new QCR is based on the technique of simultaneous diagonalization which has become one of the hottest tools in the area of quadratic programming. We first demonstrate that the 'best' QCR can be achieved by solving a Shor relaxation of the original problem. Then, we design a branch-and-bound algorithm based on the proposed QCR for obtaining the global optimal solution. Numerical experiments with extended Celis–Dennis–Tapia problem and optimal spectrum sharing problem are conducted to show that our proposed QCR well balances the bound quality and computing efficiency, hence it is very competitive with two state-of-the-art QCRs when they are executed by the same branch-and-bound scheme. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an efficient subgradient extragradient (SE) based method for solving variational inequality problems with monotone operator in Hilbert space. In many existing SE methods, two values of operator are needed over each iteration and the Lipschitz constant of the operator or linesearch is required for estimating step sizes, which are usually not practical and expensive. To overcome these drawbacks, we present an inertial SE based algorithm with adaptive step sizes, estimated by using an approximation of the local Lipschitz constant without running a linesearch. Each iteration of the method only requires a projection on the feasible set and a value of the operator. The numerical experiments illustrate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we propose a hybrid conjugate gradient method for unconstrained optimization, obtained by a convex combination of the LS and KMD conjugate gradient parameters. A favourite property of the proposed method is that the search direction satisfies the Dai–Liao conjugacy condition and the quasi-Newton direction. In addition, this property does not depend on the line search. Under a modified strong Wolfe line search, we establish the global convergence of the method. Numerical comparison using a set of 109 unconstrained optimization test problems from the CUTEst library show that the proposed method outperforms the Liu–Storey and Hager–Zhang conjugate gradient methods. [ABSTRACT FROM AUTHOR]

This paper proposes a globally convergent regularized interior point method that involves a specifically designed regularization strategy for constrained optimization. The main concept of the proposed algorithm is that when a proper regularization parameter is selected, the direction obtained from the regularized barrier equation is a descent direction for either the objective function or constraint violation. Accordingly, by embedding a flexible strategy of choosing a regularization parameter in a trust-funnel-like interior point scheme, we propose the new algorithm. Global convergence under the mild assumptions of relaxed constant rank constraint qualification (RCRCQ) and local consistency of the linearized active and equality constraints is shown. Preliminary numerical experiments are conducted, and the results are encouraging. [ABSTRACT FROM AUTHOR]