A novel method for non-probabilistic convex modelling based on data from practical engineering

In this paper, a novel method for non-probabilistic convex modelling with the bounds to precisely encircle all the data of uncertain parameters extracted from practical engineering is developed. The method is based on the traditional statistical method and the correlation analysis technique. Mean values and correlation coefficients of uncertain parameters are first calculated by utilizing the information of all the given data. Then, a simple yet effective optimization procedure is first introduced in the mathematical modelling process for uncertain parameters to obtain their precise bounds. This procedure works by optimizing the area of the convex model, at the same time, covering all the given data. Thus, the effective mathematical expression of the convex models are finally formulated. To test the prediction capability and generalization ability of the proposed convex modelling method, evaluation criteria, i.e. volume ratio, standard volume ratio, and prediction accuracy are established. The performance of the proposed method is systematically studied and compared with other existing competitive methods through test standards. The results demonstrate the effectiveness and efficiency of the present method.     

Some characterizations of robust optimal solutions for uncertain convex optimization problems

In this paper, we consider robust optimal solutions for a convex optimization problem in the face of data uncertainty both in the objective and constraints. By using the properties of the subdifferential sum formulae, we first introduce a robust-type subdifferential constraint qualification, and then obtain some completely characterizations of the robust optimal solution of this uncertain convex optimization problem. We also investigate Wolfe type robust duality between the uncertain convex optimization problem and its uncertain dual problem by proving duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem. Moreover, we show that our results encompass as special cases some optimization problems considered in the recent literature.     

Super parametric convex model and its application for non-probabilistic reliability-based design optimization

In this study, we attempt to propose a new super parametric convex model by giving the mathematical definition, in which an effective minimum volume method is constructed to give a reasonable enveloping of limited experimental samples by selecting a proper super parameter. Two novel reliability calculation algorithms, including nominal value method and advanced nominal value method, are proposed to evaluate the non-probabilistic reliability index. To investigate the influence of non-probabilistic convex model type on non-probabilistic reliability-based design optimization, an effective approach based on advanced nominal value method is further developed. Four examples, including two numerical examples and two engineering applications, are tested to demonstrate the superiority of the proposed non-probabilistic reliability analysis and optimization technique.     

Structural design optimization based on hybrid time-variant reliability measure under non-probabilistic convex uncertainties

Structural safety assessment issue, considering the influence of uncertain factors, is widely concerned currently. However, uncertain parameters present time-variant characteristics during the entire structural design procedure. Considering materials aging, loads varying and damage accumulation, the current reliability-based design optimization (RBDO) strategy that combines the static/time-invariant assumption with the random theory will be inapplicable when tackling with the optimal design issues for lifecycle mechanical problems. In light of this, a new study on non-probabilistic time-dependent reliability assessment and design under time-variant and time-invariant convex mixed variables is investigated in this paper. The hybrid reliability measure is first given by the first-passage methodology, and the solution aspects should depend on the regulation treatment and the convex theorem. To guarantee the rationality and efficiency of the optimization task, the improved GA algorithm is involved. Two numerical examples are discussed to demonstrate the validity and usage of the presented methodology.     

Turnpike theorems for convex problems

We prove turnpike theorems for systems described by differential inclusions with convex graphs.     

Extremal problems for convex polygons

Consider a convex polygon V _n with n sides, perimeter P _n , diameter D _n , area A _n , sum of distances between vertices S _n and width W _n . Minimizing or maximizing any of these quantities while fixing another defines 10 pairs of extremal polygon problems (one of which usually has a trivial solution or no solution at all). We survey research on these problems, which uses geometrical reasoning increasingly complemented by global optimization methods. Numerous open problems are mentioned, as well as series of test problems for global optimization and non-linear programming codes.     

Minimax principles for convex eigenvalue problems

Positive solutions of the nonlinear eigenvalue problem Au = σu for a forced, convex, isotone, compact operator on a partially ordered locally convex topological vector space E are considered. Denote by $\text{σ}{}^1$ the infimum of the set of σ for which the equation has a solution u in the positive cone K of E. $\text{σ}{}^1$ is characterized as the saddle value of a functional J_A determined by A and defined on the Cartesian product of K and its dual $\text{K}{}^1$.     

Random algorithms for convex minimization problems

This paper deals with iterative gradient and subgradient methods with random feasibility steps for solving constrained convex minimization problems, where the constraint set is specified as the intersection of possibly infinitely many constraint sets. Each constraint set is assumed to be given as a level set of a convex but not necessarily differentiable function. The proposed algorithms are applicable to the situation where the whole constraint set of the problem is not known in advance, but it is rather learned in time through observations. Also, the algorithms are of interest for constrained optimization problems where the constraints are known but the number of constraints is either large or not finite. We analyze the proposed algorithm for the case when the objective function is differentiable with Lipschitz gradients and the case when the objective function is not necessarily differentiable. The behavior of the algorithm is investigated both for diminishing and non-diminishing stepsize values. The almost sure convergence to an optimal solution is established for diminishing stepsize. For non-diminishing stepsize, the error bounds are established for the expected distances of the weighted averages of the iterates from the constraint set, as well as for the expected sub-optimality of the function values along the weighted averages.     

Primal-dual subgradient methods for convex problems

In this paper we present a new approach for constructing subgradient schemes for different types of nonsmooth problems with convex structure. Our methods are primal-dual since they are always able to generate a feasible approximation to the optimum of an appropriately formulated dual problem. Besides other advantages, this useful feature provides the methods with a reliable stopping criterion. The proposed schemes differ from the classical approaches (divergent series methods, mirror descent methods) by presence of two control sequences. The first sequence is responsible for aggregating the support functions in the dual space, and the second one establishes a dynamically updated scale between the primal and dual spaces. This additional flexibility allows to guarantee a boundedness of the sequence of primal test points even in the case of unbounded feasible set (however, we always assume the uniform boundedness of subgradients). We present the variants of subgradient schemes for nonsmooth convex minimization, minimax problems, saddle point problems, variational inequalities, and stochastic optimization. In all situations our methods are proved to be optimal from the view point of worst-case black-box lower complexity bounds.     

Bounds on stochastic convex allocation problems

We consider a stochastic convex program arising in a certain resource allocation problem. The uncertainty is in the demand for a resource which is to be allocated among several competing activities under convex inventory holding and shortage costs. The problem is cast as a two–period stochastic convex program and we derive tight upper and lower bounds to the problem using marginal distributions of the demands, which may be stochastically dependent. It turns out that these bounds are tighter than the usual bounds in the literature which are based on limited moment information of the underlying random variables. Numerical examples illustrate the bounds.     

Relaxation techniques for strictly convex network problems

Gauss—Seidel type relaxation techniques are applied in the context of strictly convex pure networks with separable cost functions. The algorithm is an extension of the Bertsekas—Tseng approach for solving the linear network problem and its dual as a pair of monotropic programming problems. The method is extended to cover the class of generalized network problems. Alternative internal tactics for the dual problem are examined. Computational experiments — aimed at the improved efficiency of the algorithm — are presented.This research was supported in part by National Science Foundation Grant No. DCR-8401098-A01.     

Global error bound for convex inclusion problems

The existence of global error bound for convex inclusion problems is discussed in this paper, including pointwise global error bound and uniform global error bound. The existence of uniform global error bound has been carefully studied in Burke and Tseng (SIAM J. Optim. 6(2), 265–282, 1996) which unifies and extends many existing results. Our results on the uniform global error bound (see Theorem 3.2) generalize Theorem 9 in Burke and Tseng (1996) by weakening the constraint qualification and by widening the varying range of the parameter. As an application, the existence of global error bound for convex multifunctions is also discussed.     

Restricted simplicial decomposition for convex constrained problems

The strategy of Restricted Simplicial Decomposition is extended to convex programs with convex constraints. The resulting algorithm can also be viewed as an extension of the (scaled) Topkis—Veinott method of feasible directions in which the master problem involves optimization over a simplex rather than the usual line search. Global convergence of the method is proven and conditions are given under which the master problem will be solved a finite number of times. Computational testing with dense quadratic problems confirms that the method dramatically improves the Topkis—Veinott algorithm and that it is competitive with the generalized reduced gradient method.This research was supported in part by NSF Grants ECS-8516365 and DDM-8814075.     

Well-posedness for convex symmetric vector quasi-equilibrium problems

In this paper, the notion of a generalized Levitin–Polyak well-posedness is defined for symmetric vector quasi-equilibrium problems. Sufficient conditions are given for the generalized Levitin–Polyak well-posedness. Moreover, it is shown that the results can be refined in the convex case.     

Relaxation techniques for strictly convex network problems

Gauss—Seidel type relaxation techniques are applied in the context of strictly convex pure networks with separable cost functions. The algorithm is an extension of the Bertsekas—Tseng approach for solving the linear network problem and its dual as a pair of monotropic programming problems. The method is extended to cover the class of generalized network problems. Alternative internal tactics for the dual problem are examined. Computational experiments —aimed at the improved efficiency of the algorithm — are presented. This research was supported in part by National Science Foundation Grant No. DCR-8401098-A0l.     

Rigorous convex underestimators for general twice-differentiable problems

In order to generate valid convex lower bounding problems for nonconvex twice-differentiable optimization problems, a method that is based on second-order information of general twice-differentiable functions is presented. Using interval Hessian matrices, valid lower bounds on the eigenvalues of such functions are obtained and used in constructing convex underestimators. By solving several nonlinear example problems, it is shown that the lower bounds are sufficiently tight to ensure satisfactory convergence of the BB, a branch and bound algorithm which relies on this underestimation procedure [3].     

Piece adding technique for convex maximization problems

In this article we provide an algorithm, where to escape from a local maximum y of convex function f over D, we (locally) solve piecewise convex maximization max{min{f (x) − f (y), p _y (x)} | x ∈ D} with an additional convex function p _y (·). The last problem can be seen as a strictly convex improvement of the standard cutting plane technique for convex maximization. We report some computational results, that show the algorithm efficiency.     

Regularized optimization methods for convex MINLP problems

We propose regularized cutting-plane methods for solving mixed-integer nonlinear programming problems with nonsmooth convex objective and constraint functions. The given methods iteratively search for trial points in certain localizer sets, constructed by employing linearizations of the involved functions. New trial points can be chosen in several ways; for instance, by minimizing a regularized cutting-plane model if functions are costly. When dealing with hard-to-evaluate functions, the goal is to solve the optimization problem by performing as few function evaluations as possible. Numerical experiments comparing the proposed algorithms with classical methods in this area show the effectiveness of our approach.     

-Optimality conditions for composed convex optimization problems

The aim of the present paper is to provide a formula for the -subdifferential of f+gh different from the ones which can be found in the existent literature. Further we equivalently characterize this formula by using a so-called closedness type regularity condition expressed by means of the epigraphs of the conjugates of the functions involved. Even more, using the -subdifferential formula we are able to derive necessary and sufficient conditions for the -optimal solutions of composed convex optimization problems.