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The analysis of the principal-agent problem usually requires the classical first-order approach (FOA). However, the validity of the FOA makes restrictive assumptions on the problem under consideration such as the convexity of the distribution function condition. The main aim of this paper is to compute the optimal wages and characterize a closed form solution to the risk-neutral principal-agent problem with limited liability constraints. The development in this paper mainly invokes certain techniques in the semi-infinite programming rather than the FOA. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new inexact dual decomposition algorithm for solving separable convex optimization problems. This algorithm is a combination of three techniques: dual Lagrangian decomposition, smoothing and excessive gap. The algorithm has low computational complexity since it consists in only one primal step and two dual steps at each iteration and allows one to solve the subproblem of each component inexactly and in parallel. Moreover, the algorithmic parameters are updated automatically without any tuning strategy as it happens in augmented Lagrangian approaches. We analyse the convergence of the algorithm and estimate itsanalytical worst-case complexity for both the primal–dual suboptimality and the primal feasibility violation, whereis a given accuracy. Extensive numerical tests confirm that our method is numerically more efficient than the classical decomposition methods from the literature. [ABSTRACT FROM AUTHOR]

In this paper, a new nonlinear scalarization function, which is a generalization of the oriented distance function, is introduced. Some properties of the function are discussed. Then the function is applied to obtain some new optimality conditions and scalar representations for set-valued vector optimization problems with set optimization criteria. In terms of the function and the image space analysis, some new alternative results for generalized parametric systems are derived. [ABSTRACT FROM AUTHOR]

We study semi-convex frontier (SCF) optimization problems where objective functions can be semi-convex and constraint sets can be non-polyhedron, which stem from a growing range of optimization applications such as frontier analysis, multi-objective programming in economics. The new findings of this paper can be summarized as follows: (1) We characterize non-dominated points of a non-polyhedron optimal solution set of a semi-convex frontier program. (2) We obtain optimality conditions of a constant modulus SCF program, of which the objective function is semi-convex with a constant semiconvexity modulus. (3) We obtain a non-smooth Hölder stability of the optimal solutions of a semiconvex frontier program. (4) We use generalized differentiability to establish sensitivity analysis of the optimal value function of a semi-convex frontier program. [ABSTRACT FROM AUTHOR]

Concave gauge functions were introduced to give an analytical representation of cones. In particular, they give a simple and a practical representation of the positive orthant. There is a wide choice of concave gauge functions with interesting properties, representing the same cone. Besides the fact that a concave gauge cannot be identically zero on a cone(), it may be continuous, differentiable and evenon its interior. The purpose of the present paper is to present another approach to penalizing the positivity constraints of a linear programme using an arbitrary strictly quasi-concave gauge representation. Throughout the paper, we generalize the concept of the central path and the analytic centre in terms of these gauges, introduce the differential barrier concept and establish its relationship with strict quasi-concavity. [ABSTRACT FROM AUTHOR]

Inverse data envelopment analysis (InDEA) is a well-known approach for short-term forecasting of a given decision-making unit (DMU). The conventional InDEA models use the production possibility set (PPS) that is composed of an evaluated DMU with current inputs and outputs. In this paper, we replace the fluctuated DMU with a modified DMU involving renewal inputs and outputs in the PPS since the DMU with current data cannot be allowed to establish the new PPS. Besides, the classical DEA models such as InDEA are assumed to consider perfect knowledge of the input and output values but in numerous situations, this assumption may not be realistic. The observed values of the data in these situations can sometimes be defined as interval numbers instead of crisp numbers. Here, we extend the InDEA model to interval data for evaluating the relative efficiency of DMUs. The proposed models determine the lower and upper bounds of the inputs of a given DMU separately when its interval outputs are changed in the performance analysis process. We aim to remain the current interval efficiency of a considered DMU and the interval efficiencies of the remaining DMUs fixed or even improve compared with the current interval efficiencies. [ABSTRACT FROM AUTHOR]

In this paper, existence of critical point and weak efficient point of vector optimization problem is studied. A sequence of points in n-dimension is generated using positive definite matrices like Quasi-Newton method. It is proved that accumulation points of this sequence are critical points or weak efficient points under different conditions. An algorithm is provided in this context. This method is free from any kind of priori chosen weighting factors or any other form of a priori ranking or ordering information for objective functions. Also, this method does not depend upon initial point. The algorithm is verified in numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we establish new necessary conditions for-minimizers of vector optimization problems with and without geometric constraints with respect tononconvex and nonsolid ordering sets. We use the dual-space approach based on advanced tools of variational analysis and generalized differentiation. It allows us to handle vector optimization problems without converting them to equivalent problems of scalar optimization. They are new even in vector optimization for ordering cones. [ABSTRACT FROM AUTHOR]

The Semi-Infinite Simplex Algorithm (SISA) with applications to so called Semi-Infinite Linear Programs (SILP) with capacity constraints as a direct analog to column generating techniques in linear programming methods ofwas presented by the author in 1978. Later in 1981, the SISA for SILP in partially ordered spaces was described with more algebraic framework in terms of feasible basic solutions, extreme points and so on. Now this paper has two aims: to reflect the history of one branch of the so-called Leipzig-Optimization-Tree with roots J. Focke/A. Göpfert / R. Klötzler and to demonstrate some applications of the Simplex Algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we first gather existence results for linear and for pseudo-monotone variational inequalities in reflexive Banach spaces. We discuss the necessity of the involved coerciveness conditions and their relationship. Then, we combine Mosco convergence of convex closed sets with an approximation of pseudo-monotone bifunctions and provide a convergent approximation procedure for pseudo-monotone variational inequalities in reflexive Banach spaces. Since hemivariational inequalities in linear elasticity are pseudo-monotone, our approximation method applies to nonmonotone contact problems. We sketch how regularization of the involved nonsmooth functionals together with finite element approximation lead to an efficient numerical solution method for these nonconvex nondifferentiable optimization problems. To illustrate our theory, we give a numerical example of a 2D linear elastic block under a given nonmonotone contact law. [ABSTRACT FROM AUTHOR]

In this paper a problem of air pollution control is studied, posing it as a multi-objective control problem of partial differential equations. The original problem, dealing with the optimal management of a set of industrial plants inside a populated area, is formulated by means of the diffusion transport equation, including a linear reaction term and source terms modelled by Dirac deltas. Introducing adjoint state techniques, the problem transforms into a problem of multi-objective optimization in Banach spaces, where the large number of objective functions discourages the complete search of its Pareto front. Therefore, in order to solve the problem, two interactive methods of multi-objective programming are proposed: the VIA and the STEM algorithms. Finally, the paper illustrates how to combine both algorithms to solve in a more effective way a realistic problem posed in the Metropolitan Area of Guadalajara (Mexico). [ABSTRACT FROM AUTHOR]

In this paper, we discuss the stability of the sets of (weak) minimal points and (weak) efficient points of vector optimization problems. Assuming that the objective functions are (strictly) properly quasi convex, and the data ofthe approximate problems converges to the data of the original problems in the sense of Painlevé–Kuratowski, we establish the Painlevé–Kuratowski set convergence of the sets of (weak) minimal points and (weak) efficient points of the approximate problems to the corresponding ones of original problem. Our main results improve and extend the results of the recent papers. [ABSTRACT FROM PUBLISHER]

In the present paper, a solution approach to fuzzy bilevel optimization problem is given for the linear case. We use-cuts of the fuzzy polytopes to attack the fuzzy optimization problem on upper level. Then, crisp optimization problems arise. Thus, the solution of the fuzzy upper level problem turns out to be a union of the solutions of these crisp problems on all-cuts. The solution approach for the lower level fuzzy optimization problem was investigated by the author in previous publications and is implemented in the present paper. [ABSTRACT FROM PUBLISHER]

The purpose of this paper is to present two new iterative algorithms to find the minimum norm fixed point of nonexpansive nonself-mappings in the framework of Hilbert spaces. The results presented in this paper improve and extend the corresponding ones announced by Yao and Xu [Yao Y, Xu HK. Iterative methods for finding minimum norm fixed point of mappings with applications. Optimization. 2011;60:645–658] and many others. Some applications to convex minimization and split feast problems(SFP) are also included. [ABSTRACT FROM PUBLISHER]

In the present paper, an optimal solution of a nondifferentiable fuzzy optimization problem is characterized to be fuzzy. We use the-cut to attack the fuzzy optimization problem. Then, a bicriterial optimization problem arises. Thus, the solution of the fuzzy optimization problem turns out to be Pareto optimal for the bicriterial optimization problem. This approach is used to derive necessary and sufficient optimality conditions for the optimal solution of the nondifferentiable fuzzy optimization problem. [ABSTRACT FROM AUTHOR]

This paper conducts variational analysis of circular programs, which form a new class of optimization problems in nonsymmetric conic programming, important for optimization theory and its applications. First, we derive explicit formulas in terms of the initial problem data to calculate various generalized derivatives/co-derivatives of the projection operator associated with the circular cone. Then we apply generalized differentiation and other tools of variational analysis to establish complete characterizations of full and tilt stability of locally optimal solutions to parameterized circular programs. [ABSTRACT FROM AUTHOR]

The aim of this paper is, in the setting of normed spaces with a cone K non necessarily solid, to study new relations among set scalarization functions that are extensions of the oriented distance of Hiriart-Urruty. Moreover, we deal with a set scalarization function of sup-inf type, we investigate its relation to the cone-properness and cone-boundedness and it is related to other set scalarizations existing in the literature. In particular, with the norm induced by the Minkowski's functional, we obtain relations with a set scalarization which is an extension of the so called Gerstewitz's scalarization function. [ABSTRACT FROM AUTHOR]

We study and solve the two-stage stochastic extended second-order cone programming problem. We show that the barrier recourse functions and the composite barrier functions for this optimization problem are self-concordant families with respect to barrier parameters. These results are used to develop primal decomposition-based interior-point algorithms. The worst case iteration complexity of the developed algorithms is shown to be the same as that for the short- and long-step primal interior algorithms applied to the extensive formulation of our problem. [ABSTRACT FROM AUTHOR]

Artificial Bee Colony (ABC) optimization algorithm is a swarm intelligence-based nature inspired algorithm, which has been proved a competitive algorithm with some popular nature-inspired algorithms. ABC has been found to be more efficient in exploration as compared to exploitation. With a motivation to balance exploration and exploitation capabilities of ABC, this paper presents an adaptive version of ABC. In this adaptive version, step size in solution modification and ABC parameter ‘limit’ are set adaptively based on current fitness values. In the present self-adaptive ABC, good solutions are appointed to exploit the search region in their neighbourhood, while worse solutions are appointed to explore the search region. The better solutions are given higher chances to update themselves with the help of parameter ‘limit’, which changes adaptively in the present study. The experiments on 16 unbiased test problems of different complexities show that the proposed strategy outperforms the basic ABC and some recent variants of ABC. [ABSTRACT FROM AUTHOR]

In this paper, we consider quantitative optimization problems on decomposable discrete systems. We restrict ourselves to labeled trees as the description of the systems and we use weighted automata on them as our computational model. We introduce a new kind of labeled decomposable trees,sum-like weighted labeled trees, and propose a method, which allows us to reduce the solution of an optimization problem, defined in a fragment of Weighted Monadic Second Order Logic, on such a tree to the solution of effectively derived problems, defined in the same logic, on its components with some additional post-processing. The approach originates from a method, proposed first by Feferman and Vaught in 1959, which we adapt and generalize. We adapt the method to the case of (fragments of) Weighted Monadic Second Order Logic and generalize it to the case of sum-like labeled trees rather than disjoint union and product. The main result of the paper may be applied in the wide range of optimization problems, such as critical path analysis or project planning, network optimization, generalized assignment problem, routing and scheduling as well as in the modern document languages like XML, image processing and compression, probabilistic systems or speech-to-text processing. [ABSTRACT FROM PUBLISHER]

Information about several papers discusses in 11th International Conference on Stochastic Programming which was held in Vienna, Austria on August 27, 2007 is presented. The conference was aimed at closing the gap between the field of stochastic programming and the area of robust optimization and optimization under uncertainty. The conference featured Arkadi Nemirovski, Darinka Dentcheva and Andrzej Ruszczynski.

This paper introduces the Fejér-monotone hybrid steepest descent method (FM-HSDM), a new member to the HSDM family of algorithms, for solving affinely constrained minimization tasks in real Hilbert spaces, where convex smooth and non-smooth losses compose the objective function. FM-HSDM offers sequences of estimates which converge weakly and, under certain hypotheses, strongly to solutions of the task at hand. In contrast to its HSDM's precursors, FM-HSDM enjoys Fejér monotonicity, the step-size parameter stays constant across iterations to promote convergence speed-ups of the sequence of estimates to a minimizer, while only Lipschitzian continuity, and not strong monotonicity, of the derivative of the smooth-loss function is needed to ensure convergence. FM-HSDM utilizes fixed-point theory, variational inequalities and affine-nonexpansive mappings to accommodate affine constraints in a more versatile way than state-of-the-art primal-dual techniques and the alternating direction method of multipliers do. Recursions can be tuned to score low computational footprints, well-suited for large-scale optimization tasks, without compromising convergence guarantees. Results on the rate of convergence to an optimal point are also presented. Finally, numerical tests on synthetic data are used to validate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we present two penalty-splitting inspired iteration schemes (PFFSA) and (PFFSA) for hierarchical equilibrium problems in Hilbert space. Based on the Opial-Passty lemma, we propose weak ergodic convergence and weak convergence of the iterative sequences generated by the Forward-Forward algorithm (PFFSA) and the Forward-Backward algorithm (PFFSA), which are proved under quite mild conditions: the bifunction of the two level equilibrium problems are supposed pseudomonotone. For strong convergence, we first add a strong monotonicity condition on the objective bifunction. We present after, a strong convergence result of algorithm (PFFSA) by adding a topological assumption, i.e. the objective bifunction is of class . Some examples are given to illustrate our results. The first example deals with pseudomonotone variational inequalities and convex minimization problem in the upper level problem. In the second one, we propose a convex minimization in the lower-level problem, where strong convergence of (PFFSA) to a minimum point is insured under infcompactness condition for objective function. These convergence results are new and generalize some recent results in this field. [ABSTRACT FROM AUTHOR]

Optimization over the efficient set of a multi-objective optimization problem is a mathematical model for the problem of selecting a most preferred solution that arises in multiple criteria decision-making to account for trade-offs between objectives within the set of efficient solutions. In this paper, we consider a particular case of this problem, namely that of optimizing a linear function over the image of the efficient set in objective space of a convex multi-objective optimization problem. We present both primal and dual algorithms for this task. The algorithms are based on recent algorithms for solving convex multi-objective optimization problems in objective space with suitable modifications to exploit specific properties of the problem of optimization over the efficient set. We first present the algorithms for the case that the underlying problem is a multi-objective linear programme. We then extend them to be able to solve problems with an underlying convex multi-objective optimization problem. We compare the new algorithms with several state of the art algorithms from the literature on a set of randomly generated instances to demonstrate that they are considerably faster than the competitors. [ABSTRACT FROM AUTHOR]

In this paper, we propose a large-update primal-dual interior-point algorithm for linear optimization problems based on a new kernel function with a trigonometric growth term. By simple analysis, we prove that in the large neighbourhood of the central path, the worst case iteration complexity of the new algorithm is bounded above by , which matches the currently best known iteration bound for large-update methods. Moreover, we show that, most of the so far proposed kernel functions can be rewritten as a kernel function with trigonometric growth term. Finally, numerical experiments on some test problems confirm that the new kernel function is well promising in practice in comparison with some existing kernel functions in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose a nonsmooth trust region algorithm for nonconvex optimization problems. The algorithm is based on notion of the Goldstein -subdifferential, which are subgradients computed in some neighbourhoods of a point. The proposed method contains a new quadratic model of the classical trust region method, in which the gradient vector is replaced by a quasisecant. Then we apply a combined approach based on the Cauchy point and the dog-leg methods in order to solve the obtained model. The global convergence is established under some suitable assumptions. Finally, the algorithm is implemented in the MATLAB environment and applied on some nonsmooth test problems. Numerical results on some small-scale and large-scale nonsmooth optimization test problems illustrate the efficiency of the proposed algorithm in the practical computation. [ABSTRACT FROM AUTHOR]

In this paper, based on a new smoothing function, the general box constrained variational inequalities are solved by a smoothing Newton algorithm with a nonmonotone line search. The proposed algorithm is proved to be globally and locally superlinearly convergent under suitable assumptions. The preliminary numerical results are reported. [ABSTRACT FROM AUTHOR]

In the paper, we consider the bioprocess system optimal control problem. Generally speaking, it is very difficult to solve this problem analytically. To obtain the numerical solution, the problem is transformed into a parameter optimization problem with some variable bounds, which can be efficiently solved using any conventional optimization algorithms, e.g. the improved Broyden-Fletcher-Goldfarb-Shanno algorithm. However, in spite of the improved Broyden-Fletcher-Goldfarb-Shanno algorithm is very efficient for local search, the solution obtained is usually a local extremum for non-convex optimal control problems. In order to escape from the local extremum, we develop a novel stochastic search method. By performing a large amount of numerical experiments, we find that the novel stochastic search method is excellent in exploration, while bad in exploitation. In order to improve the exploitation, we propose a hybrid numerical optimization algorithm to solve the problem based on the novel stochastic search method and the improved Broyden-Fletcher-Goldfarb-Shanno algorithm. Convergence results indicate that any global optimal solution of the approximate problem is also a global optimal solution of the original problem. Finally, two bioprocess system optimal control problems illustrate that the hybrid numerical optimization algorithm proposed by us is low time-consuming and obtains a better cost function value than the existing approaches. [ABSTRACT FROM AUTHOR]

In this paper, we study different classes of generalized convex/quasiconvex set-valued maps, defined by means of the l-type and u-type preorder relations, currently used in set-valued optimization. In particular, we identify those classes of set-valued maps for which it is possible to extend the classical characterization of convex real-valued functions by quasiconvexity of their affine perturbations. [ABSTRACT FROM AUTHOR]

In this paper, we follow Kuroiwa’s set approach in set optimization, which proposes to compare values of a set-valued objective map F with respect to various set order relations. We introduce a Hausdorff-type distance relative to an ordering cone between two sets in a Banach space and use it to define a directional derivative for F. We show that the distance has nice properties regarding set order relations and the directional derivative enjoys most properties of the one of a scalar single-valued function. These properties allow us to derive necessary and/or sufficient conditions for various types of maximizers and minimizers of F. [ABSTRACT FROM AUTHOR]

Scalarization in vector optimization is often closely connected to the minimization of Gerstewitz functionals. In this paper, the minimizer sets of Gerstewitz functionals are investigated. Conditions are given under which such a set is nonempty and compact. Interdependencies between solutions of problems with different parameters or with different feasible point sets are shown. Consequences for the parameter control in scalarization methods are proved. It is pointed out that the minimization of Gerstewitz functionals is equivalent to an optimization problem which generalizes the scalarization by Pascoletti and Serafini. The results contain statements about minimizers of certain Minkowski functionals and norms. Some existence results for solutions of vector optimization problems are derived. [ABSTRACT FROM AUTHOR]

Typically, exact information of the whole subdifferential is not available for intrinsically nonsmooth objective functions such as for marginal functions. Therefore, the semismoothness of the objective function cannot be proved or is even violated. In particular, in these cases standard nonsmooth methods cannot be used. In this paper, we propose a new approach to develop a converging descent method for this class of nonsmooth functions. This approach is based on continuous outer subdifferentials introduced by us. Further, we introduce on this basis a conceptual optimization algorithm and prove its global convergence. This leads to a constructive approach enabling us to create a converging descentmethod. Within the algorithmic framework, neither semismoothness nor calculation of exact subgradients are required. This is in contrast to other approaches which are usually based on the assumption of semismoothness of the objective function. [ABSTRACT FROM AUTHOR]

In this paper, we consider the robust mean variance optimization problem where the probability distribution of assets’ returns is multivariate normal and the uncertain mean and covariance are controlled by a constraint involving Rényi divergence. We present the closed-form solutions for the robust mean variance optimization problem and find that the choice of order parameter which is related to the Rényi divergence measure will not impact optimal portfolio strategy under the cases that the mean vector and the covariance matrix are uncertain, respectively. Moreover, we obtain the closed-form solution for the robust mean variance optimization problem under the case that the mean vector and the covariance matrix are both uncertain. We illustrate the efficiency of our results with an example. [ABSTRACT FROM PUBLISHER]

The aim of this paper is to propose a new multiple subgradient descent bundle method for solving unconstrained convex nonsmooth multiobjective optimization problems. Contrary to many existing multiobjective optimization methods, our method treats the objective functions as they are without employing a scalarization in a classical sense. The main idea of this method is to find descent directions for every objective function separately by utilizing the proximal bundle approach, and then trying to form a common descent direction for every objective function. In addition, we prove that the method is convergent and it finds weakly Pareto optimal solutions. Finally, some numerical experiments are considered. [ABSTRACT FROM AUTHOR]

The paper proposes a new shrinking gradient-like projection method for solving equilibrium problems. The algorithm combines the generalized gradient-like projection method with the monotone hybrid method. Only one optimization program is solved onto the feasible set at each iteration in our algorithm without any extra-step dealing with the feasible set. The absence of an optimization problem in the algorithm is explained by constructing slightly different cutting-halfspace in the monotone hybrid method. Theorem of strong convergence is established under standard assumptions imposed on equilibrium bifunctions. An application of the proposed algorithm to multivalued variational inequality problems (MVIP) is presented. Finally, another algorithm is introduced for MVIPs in which we only use a value of main operator at the current approximation to construct the next approximation. Some preliminary numerical experiments are implemented to illustrate the convergence and computational performance of our algorithms over others. [ABSTRACT FROM AUTHOR]

Desirability functions are increasingly used in multi-criteria decision-making which we support by modern optimization. It is necessary to formulate desirability functions to obtain a generalized version with a piecewise max type-structure for optimizing them in different areas of mathematics, operational research, management science and engineering by nonsmooth optimization approaches. This optimization problem needs to be robustified as regression models employed by the desirability functions are typically built under lack of knowledge about the underlying model. In this paper, we contribute to the theory of desirability functions by our robustification approach. We present how generalized semi-infinite programming and disjunctive optimization can be used for this purpose. We show our findings on a numerical example. The robustification of the optimization problem eventually aims at variance reduction in the optimal solutions. [ABSTRACT FROM AUTHOR]

In the paper by Jeyakumar et al. (J. Global Optim. (2006) 36: pp. 127-137), a sequential Lagrange multiplier condition for exact optimal solutions of a general convex programming was obtained, and as a byproduct, a new constraint qualification for an exact optimal solution of the convex programming problem to satisfy the standard Lagrange multiplier condition was also obtained. In this article, we extend these results to ε-optimal solutions of the convex programming problem. The new constraint qualification for ε-optimal solutions to satisfy a Lagrange multiplier condition is therefore weaker than the Slater-type condition proposed in the paper by Strofiot et al. (Math. Program. (1983) 25: pp. 307-328). [ABSTRACT FROM AUTHOR]

We give a direct proof of the fundamental minimization theorem for CVaR presented by Rockafellar and Uryasev in their definitive paper (Conditional value-at-risk for general loss distributions. Journal of Banking and Finance, 26, 1443-1471, 2002). [ABSTRACT FROM AUTHOR]

In this paper we examine the structure of extremals of variational problems with continuous integrands f: Rn × Rn → R1 which belong to a complete metric space of functions. Our results deal with the turnpike properties of variational problems. To have this property means that the solutions of the problems are determined mainly by the integrand, and are essentially independent of the choice of interval and endpoint conditions. [ABSTRACT FROM AUTHOR]

This paper investigates properties of the optimality equation and optimal policies in discrete time Markov decision processes with expected discounted total rewards under weak conditions that the model is well defined and the optimality equation is true. The optimal value function is characterized as a solution of the optimality equation and the structure of optimal policies is also given. [ABSTRACT FROM AUTHOR]

Set-valued optimization problems are important and fascinating field of optimization theory and widely applied to image processing, viability theory, optimal control and mathematical economics. There are two types of criteria of solutions for the set-valued optimization problems: the vector criterion and the set criterion. In this paper, we adopt the set criterion to study the optimality conditions of constrained set-valued optimization problems. We first present some characterizations of various set order relations using the classical oriented distance function without involving the nonempty interior assumption on the ordered cones. Then using the characterizations of set order relations, necessary and sufficient conditions are derived for four types of optimal solutions of constrained set optimization problem with respect to the set order relations. Finally, the image space analysis is employed to study thec-optimal solution of constrained set optimization problems, and then optimality conditions and an alternative result for the constrained set optimization problem are established by the classical oriented distance function. [ABSTRACT FROM PUBLISHER]

In this paper, we propose an iteratively reweightedminimization algorithm (IRL1 algorithm) for solving the elasticminimization problem. We prove that any sequence generated by the IRL1 algorithm is bounded and asymptotically regular. We also prove that the sequence is convergent for any rationaland the limit is a stationary point of the elasticminimization problem. Moreover, under certain conditions, we present an error bound between the limit point of convergent sequence and the sparse solution of underdetermined linear system. Numerical experiments on sparse vector recovery are presented to demonstrate the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this article, we develop a general theory of exact parametric penalty functions for constrained optimization problems. The main advantage of the method of parametric penalty functions is the fact that a parametric penalty function can be both smooth and exact unlike the standard (i.e. non-parametric) exact penalty functions that are always nonsmooth. We obtain several necessary and/or sufficient conditions for the exactness of parametric penalty functions, and for the zero duality gap property to hold true for these functions. We also prove some convergence results for the method of parametric penalty functions, and derive necessary and sufficient conditions for a parametric penalty function to not have any stationary points outside the set of feasible points of the constrained optimization problem under consideration. In the second part of the paper, we apply the general theory of exact parametric penalty functions to a class of parametric penalty functions introduced by Huyer and Neumaier, and to smoothing approximations of nonsmooth exact penalty functions. The general approach adopted in this article allowed us to unify and significantly sharpen many existing results on parametric penalty functions. [ABSTRACT FROM AUTHOR]

Proximal point method is one of the most influential procedure in solving nonlinear variational problems. It has recently been introduced in Hadamard spaces for solving convex optimization, and later for variational inequalities. In this paper, we study the general proximal point method for finding a zero point of a maximal monotone set-valued vector field defined on a Hadamard space and valued in its dual. We also give the relation between the maximality and Minty’s surjectivity condition, which is essential for the proximal point method to be well-defined. By exploring the properties of monotonicity and the surjectivity condition, we were able to show under mild assumptions that the proximal point method converges weakly to a zero point. Additionally, by taking into account the metric subregularity, we obtained the local strong convergence in linear and super-linear rates. [ABSTRACT FROM AUTHOR]

In this paper we consider a nonsmooth optimization problem with equality, inequality and set constraints. We propose new constraint qualifications and Kuhn–Tucker type necessary optimality conditions for this problem involving locally Lipschitz functions. The main tool of our approach is the notion of convexificators. We introduce a nonsmooth version of the Mangasarian–Fromovitz constraint qualification and show that this constraint qualification is necessary and sufficient for the Kuhn–Tucker multipliers set to be nonempty and bounded. [ABSTRACT FROM AUTHOR]

This paper deals with an optimal control problem of Bolza type for a class of parabolic equations. It consists in finding the initial datum that minimizes a cost functional, which comprises an energy term on the control and a regulation term given by a distributed cost on the state, and such that the final state lies within a prescribed distance to a given target. Beyond the particularities of the chosen problem, our aim here is to propose a new methodologybased on a spectral decomposition of the operator governing the evolution of the system, describe its implementation for this kind of problem, and finally perform some preliminary numerical experiments to assess its behaviour. [ABSTRACT FROM AUTHOR]

In this paper, we use techniques which originate from proof mining to give rates of asymptotic regularity and metastability for a sequence associated to the composition of two firmly nonexpansive mappings. [ABSTRACT FROM AUTHOR]

This paper provides characterizations of the weakly minimal elements of vector optimization problems and the global minima of scalar optimization problems posed on locally convex spaces whose objective functions are deterministic while the uncertain constraints are treated under the robust (or risk-averse) approach, i.e. requiring the feasibility of the decisions to be taken for any possible scenario. To get these optimality conditions we provide Farkas-type results characterizing the inclusion of the robust feasible set into the solution set of some system involving the objective function and possibly uncertain parameters. In the particular case of scalar convex optimization problems, we characterize the optimality conditions in terms of the convexity and closedness of an associated set regarding a suitable point. [ABSTRACT FROM AUTHOR]

By a scalarization method and properties of semi-algebraic sets, it is proved that both the Pareto solution set and the weak Pareto solution set of a vector variational inequality, where the constraint set is polyhedral convex and the basic operators are given by polynomial functions, have finitely many connected components. Consequences of the results for vector optimization problems are discussed in details. The results of this paper solve in the affirmative some open questions for the case of general problems without requiring monotonicity of the operators involved. [ABSTRACT FROM AUTHOR]