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We propose an inexact nonmonotone successive quadratic programming (SQP) algorithm for solving nonlinear programming problems with equality constraints and bounded variables. Regarding the value of...

In this paper, we consider gradient methods for minimizing smooth convex functions, which employ the information obtained at the previous iterations in order to accelerate the convergence towards the optimal solution. This information is used in the form of a piece-wise linear model of the objective function, which provides us with much better prediction abilities as compared with the standard linear model. To the best of our knowledge, this approach was never really applied in Convex Minimization to differentiable functions in view of the high complexity of the corresponding auxiliary problems. However, we show that all necessary computations can be done very efficiently. Consequently, we get new optimization methods, which are better than the usual Gradient Methods both in the number of oracle calls and in the computational time. Our theoretical conclusions are confirmed by preliminary computational experiments., Comment: This is an Accepted Manuscript of an article published by Taylor \& Francis in Optimization Methods and Software on 13 Jan 2021, available at https://www.tandfonline.com/doi/10.1080/10556788.2020.1858831

To solve a large-scale unconstrained optimization problem, in this paper we propose a class of spectral three-term conjugate gradient methods. We indicate that the proposed class, in fact, generate...

Newton method is one of the most powerful methods for finding solutions of nonlinear equations and for proving their existence. In its ‘pure’ form it has fast convergence near the solution, but sma...

Two transformations of gradient-descent iterative methods for solving unconstrained optimization are proposed. The first transformation is called modification and it is defined using a small enlargement of the step size in various gradient-descent methods. The second transformation is termed as hybridization and it is defined as a composition of gradient-descent methods with the Picard–Mann hybrid iterative process. As a result, several accelerated gradient-descent methods for solving unconstrained optimization problems are presented, investigated theoretically and numerically compared. The proposed methods are globally convergent for uniformly convex functions satisfying certain condition under the assumption that the step size is determined by the backtracking line search. In addition, the convergence on strictly convex quadratic functions is discussed. Numerical comparisons show better behaviour of the proposed methods with respect to some existing methods in view of the Dolan and Moré's performance profile with respect to all analysed characteristics: number of iterations, the CPU time, and the number of function evaluations.

Nonlinear gradient dynamic approach for solving the tensor complementarity problem (TCP) are presented. Theoretical analysis shows that each of the defined dynamical system models ensures the conve...

Published online: 29 Aug 2018, A number of first-order methods have been proposed for smooth multiobjective optimization for which some form of convergence to first-order criticality has been proved. Such convergence is global in the sense of being independent of the starting point. In this paper, we analyse the rate of convergence of gradient descent for smooth unconstrained multiobjective optimization, and we do it for non-convex, convex, and strongly convex vector functions. These global rates are shown to be the same as for gradient descent in single-objective optimization and correspond to appropriate worstcase complexity bounds. In the convex cases, the rates are given for implicit scalarizations of the problem vector function., Support for A.I.F. Vaz was partially provided by FCT [grant number COMPETE:POCI-01- 0145-FEDER-007043], [grant number UID/CEC/00319/2013], and support for L.N. Vicente was partially provided by FCT [grant number UID/MAT/00324/2013], [grant number P2020 SAICTPAC/0011/2015.]

4th International Conference on Computational and Experimental Science and Engineering (ICCESEN) -- OCT 04-08, 2017 -- Kemer, TURKEY, WOS: 000463781100002, In this paper, a novel sharp Augmented Lagrangian-based global optimization method is developed for solving constrained non-convex optimization problems. The algorithm consists of outer and inner loops. At each inner iteration, the discrete gradient method is applied to minimize the sharp augmented Lagrangian function. Depending on the solution found the algorithm stops or updates the dual variables in the inner loop, or updates the upper or lower bounds by going to the outer loop. The convergence results for the proposed method are presented. The performance of the method is demonstrated using a wide range of nonlinear smooth and non-smooth constrained optimization test problems from the literature., Australian Research Council's Discovery Projects funding scheme [DP140103213], The research by A.M. Bagirov was supported under Australian Research Council's Discovery Projects funding scheme (Project No. DP140103213).

In this paper, a Halpern-type proximal point algorithm for approximating a common solution of monotone inclusion problem, minimization problem (MP) and fixed point problem is introduced. Using our ...

In this work, we study the split common null point problem in the framework of Banach spaces. We propose an iterative scheme for solving the problem and then prove strong convergence theorem of the sequences generated by our iterative scheme under suitable conditions. We finally provide some numerical examples to support the main theorem.

In this paper, based on the numerical efficiency of Hestenes–Stiefel (HS) method, a new modified HS algorithm is proposed for unconstrained optimization. The new direction independent of the line search satisfies in the sufficient descent condition. Motivated by theoretical and numerical features of three-term conjugate gradient (CG) methods proposed by Narushima et al., similar to Dai and Kou approach, the new direction is computed by minimizing the distance between the CG direction and the direction of the three-term CG methods proposed by Narushima et al. Under some mild conditions, we establish global convergence of the new method for general functions when the standard Wolfe line search is used. Numerical experiments on some test problems from the CUTEst collection are given to show the efficiency of the proposed method.

It is shown how piecewise differentiable functions \(F: R^n → R^m\) that are defined by evaluation programs can be approximated locally by a piecewise linear model based on a pair of sample points x and x. We show that the discrepancy between function and model at any point x is of the bilinear order O(||x − x|| ||x − x||). This is a little surprising since x ∈ R^n may vary over the whole Euclidean space, and we utilize only two function samples F = F(x) and F = F(x), as well as the intermediates computed during their evaluation. As an application of the piecewise linearization procedure we devise a generalized Newton’s method based on successive piecewise linearization and prove for it sufficient conditions for convergence and convergence rates equaling those of semismooth Newton. We conclude with the derivation of formulas for the numerically stable implementation of the aforedeveloped piecewise linearization methods.

In this article, we consider a split common fixed point and null point problem which includes the split common fixed point problem, the split common null problem and other problems related to the fixed point problem and the null point problem. We introduce an algorithm for studying the split common fixed point and null problem for demicontractive operators and maximal monotone operators in real Hilbert spaces. We establish a strong convergence result under some suitable conditions and reduce our main result to above-mentioned problems. Moreover, we also apply our main results to the split equilibrium problem. Finally, we give numerical results to demonstrate the convergence of our algorithms.

In this paper, we propose a globally convergent BFGS method to solve Variational Inequality Problems (VIPs). In fact, a globalization technique on the basis of the hyperplane projection method is applied to the BFGS method. The technique, which is independent of any merit function, is applicable for pseudo-monotone problems. The proposed method applies the BFGS direction and tries to reduce the distance of iterates to the solution set. This property, called Fejer monotonicity of iterates with respect to the solution set, is the basis of the convergence analysis. The method applied to pseudo-monotone VIP is globally convergent in the sense that subproblems always have unique solutions, and the sequence of iterates converges to a solution to the problem without any regularity assumption. Finally, some numerical simulations are included to evaluate the efficiency of the proposed algorithm.

In this paper, we propose a new modified proximal point algorithm for a finite family of non-expansive mappings in the framework of CAT(0) spaces. We establish Δ-convergence and strong convergence theorems under some mild conditions. Our results extend some known results which appeared in the literature.

We consider dynamic optimization problems for systems described by differential-algebraic equations (DAEs). Such problems are usually solved by discretizing the full DAE. We propose techniques to symbolically eliminate many of the algebraic variables in a preprocessing step before discretization. These techniques are inspired by the causalization and tearing techniques often used when solving DAE initial value problems. Since sparsity is crucial for some dynamic optimization methods, we also propose a novel approach to preserving sparsity during this procedure.The proposed methods have been implemented in the open-source JModelica.org platform. We evaluate the performance of the methods on a suite of optimal control problems solved using direct collocation.We consider both computational time and probability of solving the problem in a timely manner. We demonstrate that the proposed methods often are an order of magnitude faster than the standard way of discretizing the full DAE, and also significantly increase probability of successful convergence. (Less)

The conjugate gradient (CG) method is one of the most popular methods for solving large-scale unconstrained optimization problems. In this paper, a new modified version of the CG formula that was introduced by Polak, Ribiere, and Polyak is proposed for problems that are bounded below and have a Lipschitz-continuous gradient. The new parameter provides global convergence properties when the strong Wolfe-Powell (SWP) line search or the weak Wolfe-Powell (WWP) line search is employed. A proof of a sufficient descent condition is provided for the SWP line search. Numerical comparisons between the proposed parameter and other recent CG modifications are made on a set of standard unconstrained optimization problems. The numerical results demonstrate the efficiency of the proposed CG parameter compared with the other CG parameters.

A nonlinear programming (NLP) model with partial orthogonality constraints, relaxed from the polynomial optimization problem, is proposed and analysed for solving sensor network localization. The NLP model is difficult to solve as the orthogonality constraints are not only non-convex but numerically expensive to preserve during iterations. To deal with this difficulty, we apply the Cayley transform (a Crank–Nicolson-like update scheme) to preserve it. Combining with the gradient descent method, we develop a curvilinear search algorithm, and analyse its convergence. In practice, we accelerate our method by taking nonlinear conjugate gradient method and Barzilai–Borwein steps. Numerical experiments are given to demonstrate the efficiency of the proposed method, for the problems with the number of sensors up to 15,000 and the distance constraints up to 135,817.

We propose a new modified proximal point algorithm combined with Halpern's iteration process for nonexpansive mappings in the framework of CAT0 spaces. We establish a strong convergence theorem under some mild conditions. Our results extend some known results which appeared in the literature.

International audience; We reexamine the work of Aupy et al. on optimal algorithms for hierarchical adjoint computations , where two levels of memories are available. The previous optimal algorithm had a quadratic execution time. Here, with structural arguments, namely periodicity, on the optimal solution, we provide an optimal algorithm in constant time and space, with appropriate pre-processing. We also provide an asymptotically optimal algorithm for the online problem, when the adjoint chain size is not known beforehand. Again, these algorithms rely on the proof that the optimal solution for hierarchical adjoint computations is weakly periodic. We conjecture a closed-form formula for the period. Finally, we assess the convergence speed of the approximation ratio for the online problem through simulations.

To save more Jacobian calculations and achieve a faster convergence rate, Yang [A higher-order Levenberg-Marquardt method for nonlinear equations, Appl. Math. Comput. 219222013, pp. 10682–10694, doi:10.1016/j.amc.2013.04.033, 65H10] proposed a higher-order Levenberg–Marquardt LM method by computing the LM step and another two approximate LM steps for nonlinear equations. Under the local error bound condition, global and local convergence of this method is proved by using trust region technique. However, it is clear that the last two approximate LM steps may be not necessarily a descent direction, and standard line search technique cannot be used directly to obtain the convergence properties of this higher-order LM method. Hence, in this paper, we employ the nonmonotone second-order Armijo line search proposed by Zhou [On the convergence of the modified Levenberg-Marquardt method with a nonmonotone second order Armijo type line search, J. Comput. Appl. Math. 239 2013, pp. 152–161] to guarantee the global convergence of this higher-order LM method. Moreover, the local convergence is also preserved under the local error bound condition. Numerical results show that the new method is efficient.

In this paper, we consider the nested decomposition method in the context of multistage stochastic problems with scenario trees where we contribute with improvements in accelerating convergence to optimum. We first propose a hybrid protocol for transmitting cuts across nodes aiming at a trade-off between the inherent benefits and drawbacks of the single aggregated versus multi-cut desegregated versions. The idea we aim to explore is to reduce the size of the problem solved at each iteration by applying the aggregated version of the cuts on the furthest nodes of the tree. We then turn our attention towards the selection of first-stage solutions by employing sampling and simulation aiming to carefully choose those first-stage solution that are potentially capable to drastically decrease the gap between the upper and lower bound at each iteration. The problems on which the latter technique is applied present a special structure, according to which the first-stage variables are linked to all further stages. We test both accelerating techniques on a generic problem which displays a blend of two-stage and multistage structure and we report significant savings in terms of CPU time and number of iterations to convergence.

In this paper we focus on the convergence analysis of the forward–backward splitting method for solving nonsmooth optimization problems in Hilbert spaces when the objective function is the sum of two convex functions. Assuming that one of the functions is Frechet differentiable and using two new linesearches, the weak convergence is established without any Lipschitz continuity assumption on the gradient. Furthermore, we obtain many complexity results of cost values at the iterates when the stepsizes are bounded below by a positive constant.

In this paper, we propose a penalty proximal alternating linearized minimization method for the large-scale sparse portfolio problems in which a sequence of penalty subproblems are solved by utilizing the proximal alternating linearized minimization framework and sparse projection techniques. For exploiting the structure of the problems and reducing the computation complexity, each penalty subproblem is solved by alternately solving two projection problems. The global convergence of the method to a Karush-Kuhn-Tucker point or a local minimizer of the problem can be proved under the characteristic of the problem. The computational results with practical problems demonstrate that our method can find the suboptimal solutions of the problems efficiently and is competitive with some other local solution methods.

In this paper, we propose a proximal partially parallel splitting method for solving convex minimization problems, where the objective function is separable into m individual operators without any coupled variables, and the structural constraint set comprises only linear functions. At each iteration of this algorithm, one selected subproblem is solved, and subsequently the remaining subproblems are solved in parallel, utilizing the new iterate information. Hence, the proposed method is a hybrid mechanism that combines the nice features of parallel decomposition methods and alternating direction methods, while simultaneously adopting the predictor–corrector strategy to ensure convergence of the algorithm. Our algorithmic framework is also amenable to admitting linearized versions of the subproblems, which frequently have closed-form solutions, thereby making the proposed method more implementable in practice. Furthermore, the worst-case convergence rate of the proposed method is obtained under both ergodic...

In this paper We consider a problem of optimal design in 2D for the wave equation with Dirichlet boundary conditions. We introduce a finite element discrete version of this problem in which the domains under consideration are polygons defined on the numerical mesh. We prove that, as the mesh size tends to zero, any limit, in the sense of the complementary-Hausdorff convergence, of discrete optimal shapes is an optimal domain for the continuous optimal design problem. We work in the functional and geometric setting introduced by V. Sverak in which the domains under consideration are assumed to have an a priori limited number of holes. We present in detail a numerical algorithm and show the efficiency of the method through various numerical experiments.

In this paper, a constraint set swelling homotopy CSSH algorithm for solving the single-level non-convex programming problem with designing piecewise linear contractual function which is equivalent to the principal-agent model with integral operator is proposed, and the existence and global convergence is proven under some mild conditions. As a comparison, a piecewise constant contract is also designed for solving the single-level non-convex programming problem with the corresponding discrete distributions. And some numerical tests are done by the proposed homotopy algorithm as well as by using fmincon in Matlab, LOQO and MINOS. The numerical results show that the CSSH algorithm is robust, feasible and effective.

In this work, we present a new class of diagonal quasi-Newton methods for solving large-scale unconstrained optimization problems. The methods are derived by means of variational principle under the generalized Frobenius norm. We show global convergence of our methods under the standard line search with Armijo condition. Numerical results are carried out in standard test problems and clearly indicate vast superiority over some classical conjugate gradient methods.

This paper proposes an incremental subgradient method for solving the problem of minimizing the sum of nondifferentiable, convex objective functions over the intersection of fixed point sets of nonexpansive mappings in a real Hilbert space. The proposed algorithm can work in nonsmooth optimization over constraint sets onto which projections cannot be always implemented, whereas the conventional incremental subgradient method can be applied only when a constraint set is simple in the sense that the projection onto it can be easily implemented. We first study its convergence for a constant step size. The analysis indicates that there is a possibility that the algorithm with a small constant step size approximates a solution to the problem. Next, we study its convergence for a diminishing step size and show that there exists a subsequence of the sequence generated by the algorithm which weakly converges to a solution to the problem. Moreover, we show the whole sequence generated by the algorithm with a diminishing step size strongly converges to the solution to the problem under certain assumptions. We also give examples of real applied problems which satisfy the assumptions in the convergence theorems and numerical examples to support the convergence analyses.

The present study aims to carefully discuss the importance of differentiability checks during the execution of methods based on gradient sampling. We stress the significance of this procedure not only from the theoretical perspective, but also in the practical implementation. We support our claims exhibiting illustrative examples where the absence of the differentiability check in the method prevents the achievement of the minimization problem solution. As possible alternatives, this manuscript presents two procedures that suppress the differentiability check without affecting the convergence of the method both in theory and in practice. Lastly, by solving a difficult control problem, we show that besides the theoretical appeal our changes may also be useful to address real problems.

This paper considers the numerical solution of linear generalized Nash equilibrium problems LGNEPs. Since many methods for nonlinear problems require the nonsingularity of some second-order derivative, standard convergence conditions are not satisfied in our linear case. We provide new convergence criteria for a potential reduction algorithm PRA that allow its application to LGNEPs. Furthermore, we discuss a projected subgradient method PSM and a penalty method that exploit some known Nikaido–Isoda function-based constrained and unconstrained optimization reformulations of the LGNEP. Moreover, it is shown that normalized Nash equilibria of an LGNEP can be obtained by solving a single linear program. All proposed algorithms are tested on randomly generated instances of economic market models that are introduced and analysed in this paper and that lead to LGNEPs with shared and with non-shared constraints. It is shown that these problems have some favourable properties that can be exploited to obtain their solutions. With the PRA and in particular with the PSM we are able to compute solutions with satisfying precision even for problems with up 10,000 variables.

In this paper, we propose a new exchange method for solving convex semi-infinite programming problems SIPs. The traditional exchange method solves a sequence of finitely relaxed subproblems, that is, subproblems with finitely many constraints chosen from the original constraints. On the other hand, our exchange method solves a sequence of new subproblems, in which the traditional finite subproblems are refined by the quadratic approximation. Under mild assumptions, the refined subproblems approximate the original SIP more precisely than the traditional subproblems. Moreover, although those subproblems are still SIPs, they can be solved efficiently by reformulating them as certain optimization problems with finitely many constraints. We establish the global convergence property of the proposed algorithm. Finally, we examine the efficiency of the algorithm by some numerical experiments.

We suggest a modification of the descent splitting methods for decomposable composite optimization problems, which maintains the basic convergence properties, but enables one to reduce the computational expenses per iteration and to provide computations in a distributed manner. It consists of making coordinate-wise steps together with a special threshold control.

We investigate various stabilization strategies for improving the column generation algorithm frequently used in recent exact methods and propose a novel stabilization technique specialized for the capacitated vehicle routing problem CVRP. We start with exploring the relationship between Lagrangian relaxation and Dantzig–Wolfe DW decomposition for CVRP. The proposed technique, called analytic centre stabilization, is based on the analytic centre cutting plane method. In particular, we generate a cutting plane in the Lagrangian dual space and use it as a new column within DW decomposition for CVRP. In computational experiments, we compare the performance of standard column generation based on the Simplex method with those of the analytic centre stabilization, the bundle method, and a hybrid of the two approaches. The results show that the proposed method is quite competitive and the hybrid method has the best convergence properties among others.

We study the rate of convergence of a projected Barzilai–Borwein method, which performs the Grippo–Lampariello–Lucidi GLL non-monotone line search along the feasible direction, for convex constrained optimization. Under mild conditions, we establish sublinear convergence of the considered method for general convex functions. Moreover, we show that the rate of convergence is R-linear for strongly convex functions. We extend the convergence results to different versions of the method based on a general GLL non-monotone line search, an adaptive non-monotone line search, and a non-monotone line search that requires an average of the successive function values decreases.

In this paper, we consider the with p∈0, 1 matrix minimization for recovering the low-rank matrices. A smoothing approach for solving this non-smooth, non-Lipschitz and non-convex optimization problem is developed, in which the smoothing parameter is treated as a decision variable and a majorization method is adopted to solve the smoothing problem. The convergence theorem shows that any accumulation point of the sequence generated by the proposed approach satisfies the first-order necessary optimality condition of the problem. As an application, we use the proposed smoothing majorization method to solve the famous matrix completion problems. Numerical results indicate that our algorithm can solve the test problems efficiently.

In this paper, for finding a minimal efficient solution of nonconvex multiobjective programming, a constraint shifting combined homotopy is constructed and the global convergence is obtained under some mild conditions. The method requires that the initial point needs to be only in the shifted feasible set not necessarily in the original feasible set, and the normal cone condition need only be satisfied in the boundary of the shifted feasible set not the original constraint set. Some numerical tests are done and made comparison with a combined homotopy interior point method. The results show that our method is feasible and effective.

A modification of the limited-memory variable metric BNS method for large-scale unconstrained optimization is proposed, which consists in corrections derived from the idea of conjugate directions of the used difference vectors for better satisfaction of the previous quasi-Newton QN conditions. In comparison with [Vlcek and Luksan, A conjugate directions approach to improve the limited-memory BFGS method, Appl. Math. Comput. 219 2012, pp. 800–809], where a similar approach is used, correction vectors from more previous iterations can be applied here. For quadratic objective functions, the improvement of convergence is the best one in some sense, all stored corrected difference vectors are conjugate and the QN conditions with these vectors are satisfied. Global convergence of the algorithm is established for convex sufficiently smooth functions. Numerical experiments demonstrate the efficiency of the new method.

In this paper we propose an augmented Lagrangian trust region method for equality constrained optimization. Different from standard augmented Lagrangian methods which minimize the augmented Lagrangian function for fixed Lagrange multiplier and penalty parameter at each iteration, the proposed method tries to minimize its second-order approximation function. We propose a new strategy for adjusting the penalty parameter. With adaptive update of Lagrange multipliers, we prove the global convergence of the proposed method. Numerical results on test problems from the CUTEr collection are also reported.

In this paper, we present a modified trust region algorithm for nonlinear equations with the trust region radii converging to zero. The algorithm calculates the Jacobian after every two computations of the step. It preserves the global convergence as the traditional trust region algorithms. Moreover, it converges nearly q-cubically under the local error bound condition, which is weaker than the nonsingularity of the Jacobian at a solution. Numerical results show that the algorithm is very efficient for both singular problems and nonsingular problems.

In the last years many solution methods for equilibrium problems (EPs) have been developed. Several different monotonicity conditions have been exploited to prove convergence. The paper investigates all the relationships between them in the framework of the so-called abstract EP. The analysis is further detailed for variational inequalities and linear EPs, which include also Nash EPs with quadratic payoffs.

This paper extends the technique used in the damped BFGS method of Powell [Algorithms for nonlinear constraints that use Lagrange functions, Math. Program. 14 (1978), 224–248] to the Broyden family of quasi-Newton methods with applications to unconstrained optimization problems. Appropriate conditions on the damped technique are proposed to enforce safely the positive definiteness property for all Broyden's updates. It is shown that this technique maintains the q-superlinear convergence property of the restricted Broyden family of methods for uniformly convex functions. It also extends the global convergence property to all members of the family. Preliminary numerical results are described which show that appropriate ways for employing the proposed technique improve the performance of all members of the Broyden family of methods substantially and significantly in certain cases. They also enforce convergence of divergent quasi-Newton methods.

In this paper, we focus on the problems of minimizing the sum of two nonsmooth functions which are possibly nonconvex. These problems arise in many applications of practical interests. We present a proximal alternating linearization algorithm which alternately generates two approximate proximal points of the original objective function. It is proved that the accumulation points of iterations converge to a stationary point of the problem. Numerical experiments validate the theoretical convergence analysis and verify the implementation of the proposed algorithm.

In this paper, we present a new nonmonotone trust region method with adaptive radius which is equipped by a nonmonotone line search technique for solving unconstrained optimization problems. The proposed method combines a modified version of the Li and Fukushima's nonmonotone technique in D.H. Li and M. Fukushima [A derivative-free line search and global convergence of Broyden like method for nonlinear equations, Optim. Methods Softw.13 2000, pp. 181–201] for solving nonlinear systems with a new variant of Shi and Guo's adaptive strategy in Z.J. Shi and J.H. Guo [A new trust region methods for unconstrained optimization, J. Comput. Appl. Math. 213 2008, pp. 509–520] for updating the trust region radius. The method performs a nonmonotone Armijo-type line search whenever the trial step is rejected. Under some standard assumptions, we provide the global convergence property as well as the superlinear and quadratic convergence rates for the new method. Numerical results show the efficiency and effectiveness of the new proposed method in practice.

Sequential optimality conditions are related to stopping criteria for nonlinear programming algorithms. Local minimizers of continuous optimization problems satisfy these conditions without constraint qualifications. It is interesting to discover whether well-known optimization algorithms generate primal–dual sequences that allow one to detect that a sequential optimality condition holds. When this is the case, the algorithm stops with a ‘correct’ diagnostic of success ‘convergence’. Otherwise, closeness to a minimizer is not detected and the algorithm ignores that a satisfactory solution has been found. In this paper it will be shown that a straightforward version of the Newton–Lagrange sequential quadratic programming method fails to generate iterates for which a sequential optimality condition is satisfied. On the other hand, a Newtonian penalty–barrier Lagrangian method guarantees that the appropriate stopping criterion eventually holds.

A stretching function technique is combined with the simulated annealing SA algorithm for the global optimization problems. In the presented algorithm, the obtained local optimum information by SA search is used to build a stretching function. To speed up the convergence of SA, SA is executed iteratively on the stretching function constructed with respect to the previously found local minima instead of on the original objective function. Furthermore, a new next trial point generation scheme in SA is designed to increase the diversity of the trial points to make the introduced technique more effective. In the numerical experiments, we first investigate and compare the effectiveness and efficiency of the combination of the stretching technique with a newly designed SA SSA, with particle swarm optimization and with differential evolution algorithm, respectively, on eight typical test functions in terms of the average number of the function evaluations and its standard deviation, and the success rate. Second, we systematically compare the numerical results of SSA with the traditional SA with elitist and SA with new generation scheme on 37 benchmark problems. The numerical results show that the hybrid method is effective for global optimization.

We analyse an implicit discretization scheme for a class of linear-quadratic optimal control problems. First, we show convergence of order Oh for the optimal values of the objective function, where h is the mesh size. Under the additional assumption that the optimal control has bang–bang structure we show that the discrete and the continuous controls coincide except on a set of measure O√h.

This paper aims to study feasible Barzilai–Borwein BB-like methods for extreme symmetric eigenvalue problems. For the two-dimensional case, we establish the local superlinear convergence result of FLBB, FSBB, FABB, and FADBB algorithms. A counter-example is also given, showing that the algorithms may cycle or stop at a non-stationary point. In order to circumvent the difficulty, we propose a safeguard in choosing the stepsize. We also adopt an adaptive non-monotone line search with an improved line search to ensure the global convergence of AFBB-like methods. Numerical experiments on a set of test problems from UF Sparse Matrix Collection demonstrate that, comparing several available codes including eigs , irbleigs and jdcg , AFBB-like methods are very useful for large-scale sparse extreme symmetric eigenvalue problems.

We consider a class of stochastic Nash equilibrium problems SNEP. Under some mild conditions, we reformulate the SNEP as a stochastic mixed complementarity problem SMCP. We apply the well-known sample average approximation SAA method to solve the SMCP. We further introduce a semismooth Newton method to solve the SAA problems. The comprehensive convergence analysis is given as well. In addition, we demonstrate the proposed approach on a stochastic Nash equilibrium model in the wholesale gas–oil markets.

Alternating direction method ADM, which decomposes a large-scale original variational inequality VI problem into a series of smaller scale subproblems, is very attractive for solving a class of VI problems with a separable structure. This type of method can greatly improve the efficiency, but cannot avoid solving VI subproblems. In this paper, we propose a hybrid splitting method with variable parameters for separable VI problems. Specifically, the proposed method solves only one strongly monotone VI subproblem and a well-posed system of nonlinear equations in each iteration. The global convergence of the new method is established under some standard assumptions as those in classical ADMs. Finally, some preliminary numerical results show that the proposed method performs favourably in practice.