Discussions on non-probabilistic convex modelling for uncertain problems

Non-probabilistic convex model utilizes a convex set to quantify the uncertainty domain of uncertain-but-bounded parameters, which is very effective for structural uncertainty analysis with limited or poor-quality experimental data. To overcome the complexity and diversity of the formulations of current convex models, in this paper, a unified framework for construction of the non-probabilistic convex models is proposed. By introducing the correlation analysis technique, the mathematical expression of a convex model can be conveniently formulated once the correlation matrix of the uncertain parameters is created. More importantly, from the theoretic analysis level, an evaluation criterion for convex modelling methods is proposed, which can be regarded as a test standard for validity verification of subsequent newly proposed convex modelling methods. And from the practical application level, two model assessment indexes are proposed, by which the adaptabilities of different convex models to a specific uncertain problem with given experimental samples can be estimated. Four numerical examples are investigated to demonstrate the effectiveness of the present study.     

A semi-analytic method for calculating non-probabilistic reliability index based on interval models

This paper proves: (1) non-probabilistic reliability index of a structure exists merely at one of intersection points at which normalized failure surfaces of the structure intersects the straight lines passing not only through origin of an normalized infinite space but also through vertices of a symmetric convex polyhedron with its sym-center at the origin, and (2) the non-probabilistic reliability index equals to absolute value of the coordinate components of a particular intersection point. Based on a reduction of the feasible region, a semi-analytical method for calculating the reliability index is developed. The method proves to be simple and of practical significance, and has several advantages over the existing unconstrained multivariate nonlinear optimization approach.     

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.     

A method for convex minimization based on translated first-order approximations

We describe an algorithm for minimizing convex, not necessarily smooth, functions of several variables, based on a descent direction finding procedure that inherits some characteristics both of standard bundle method and of Wolfe’s conjugate subgradient method. This is obtained by allowing appropriate upward shifting of the affine approximations of the objective function which contribute to the classic definition of the cutting plane function. The algorithm embeds a proximity control strategy. Finite termination is proved at a point satisfying an approximate optimality condition and some numerical results are provided.     

A proximal bundle method with inexact data for convex nondifferentiable minimization

A proximal bundle method with inexact data is presented for minimizing an unconstrained nonsmooth convex function f$f$. At each iteration, only the approximate evaluations of f$f$ and its ε$\epsilon$-subgradients are required and its search directions are determined via solving quadratic programmings. Compared with the pre-existing results, the polyhedral approximation model that we offer is more precise and a new term is added into the estimation term of the descent from the model. It is shown that every cluster of the sequence of iterates generated by the proposed algorithm is an exact solution of the unconstrained minimization problem.     

A novel cutting force modelling method for cylindrical end mill

This paper proposes a new and simplified method for the calibration of cutting force coefficients and cutter runout parameters for cylindrical end milling using the instantaneous cutting forces measured instead of average ones. The calibration procedure is derived for a mechanistic cutting force model in which the cutting force coefficients are expressed as the power functions of instantaneous uncut chip thickness (IUCT). The derivations are firstly performed by establishing mathematical relationships between instantaneous cutting forces and IUCT. Then, nonlinear algorithms are proposed to solve the established nonlinear contradiction equations. The typical features of this new calibration method lie in twofold. On the one hand, all derivations are directly based on the tangential, radial and axial cutting force components transformed from the forces which are measured in the workpiece Cartesian coordinate system. This transformation makes the calibration procedure very simple and efficient. On the other hand, only a single cutting test is needed to be performed for calibrating the cutting force coefficients that are valid over a wide range of cutting conditions. The effectiveness of the proposed method in developing cutting force model is demonstrated experimentally with a series of verification cutting tests.     

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.     

A row-action method for convex programming

We present a new method for minimizing a strictly convex function subject to general convex constraints. Constraints are used one at a time, no changes are made in the constraint functions (thus the row-action nature of the algorithm) and at each iteration a subproblem is solved consisting of minimization of the objective function subject to one or two linear equations. Convergence of the algorithm is established and the method is compared with other row-action algorithms for several relevant particular cases.Corresponding author. Research of this author was partially supported by CNPq grant No. 301280/86.     

Controlling based method for modelling chaotic dynamical systems from time series

A simple method is introduced for modelling chaotic dynamical systems from the time series, based on the concept of controlling of chaos by constant bias. In this method, a modified system is constructed by including some constants (controlling constants) into the given (original) system. The system parameters and the controlling constants are determined by solving a set of implicit nonlinear simultaneous algebraic equations which is obtained from the relation connecting original and modified systems. The method is also extended to find the form of the evolution equation of the system itself. The important advantage of the method is that it needs only a minimal number of time series data and is applicable to dynamical systems of any dimension. It also works extremely well even in the presence of noise in the time series. The method is illustrated in some specific systems of both discrete and continuous cases.     

A novel method for designing S-boxes based on chaotic maps

A method for obtaining cryptographically strong 8 × 8 S-boxes based on chaotic maps is presented and the cryptographical properties such as bijection, nonlinearity, strict avalanche criterion, output bits independence criterion and equiprobable input/output XOR distribution of these S-boxes are analyzed in detail. The results of numerical analysis also show that the S-boxes proposed are of the above properties and can resist the differential attack. Furthermore, our approach is suitable for practical application in designing cryptosystem.     

A practical algorithm for decomposing polygonal domains into convex polygons by diagonals

We present algorithms for decomposing a polygon (with holes) into convex polygons by diagonals. The methods are computationally quick, and although the partitions that they produce may not have minimal cardinality they usually have a low number of convex pieces. Thus, the methods are suitable for being used when achieving a modest load on the CPU time is more important than finding optimal decompositions as, for instance, in location problems. Part of the results in this paper are from Fernández (1999), and were presented in Fernández et al. (1998). This work has been supported by the Ministry of Science and Technology of Spain under the research projects BEC2002-01026, SEJ2005-06273/ECON (J. Fernández, B. Tóth and B. Pelegrín) and TIC2003-05982-C05-03 (L. Cánovas), in part financed by the European Regional Development Fund (ERDF).     

A characterization of ill-posed data instances for convex programming

Given a data instance of a convex program, we provide a collection of conic linear systems such that the data instance is ill-posed if and only if at least one of those systems is satisfied. This collection of conic linear systems is derived from a characterization of the boundary of the set of primal and dual feasible data instances associated with the given convex program. Received: September 1998 / Accepted: August 2000?Published online October 26, 2001     

求解非光滑凸规划的一种混合束方法

提出一种求解非光滑凸规划问题的混合束方法. 该方法通过对目标函数增加迫近项, 且对可行域增加信赖域约束进行迭代, 做为迫近束方法与信赖域束方法的有机结合, 混合束方法自动在二者之间切换, 收敛性分析表明该方法具有全局收敛性. 最后的数值算例验证了算法的有效性．     

A bundle-filter method for nonsmooth convex constrained optimization

For solving nonsmooth convex constrained optimization problems, we propose an algorithm which combines the ideas of the proximal bundle methods with the filter strategy for evaluating candidate points. The resulting algorithm inherits some attractive features from both approaches. On the one hand, it allows effective control of the size of quadratic programming subproblems via the compression and aggregation techniques of proximal bundle methods. On the other hand, the filter criterion for accepting a candidate point as the new iterate is sometimes easier to satisfy than the usual descent condition in bundle methods. Some encouraging preliminary computational results are also reported.     

A weighted gram-schmidt method for convex quadratic programming

Range-space methods for convex quadratic programming improve in efficiency as the number of constraints active at the solution decreases. In this paper we describe a range-space method based upon updating a weighted Gram-Schmidt factorization of the constraints in the active set. The updating methods described are applicable to both primal and dual quadratic programming algorithms that use an active-set strategy. Many quadratic programming problems include simple bounds on all the variables as well as general linear constraints. A feature of the proposed method is that it is able to exploit the structure of simple bound constraints. This allows the method to retain efficiency when the number ofgeneral constraints active at the solution is small. Furthermore, the efficiency of the method improves as the number of active bound constraints increases. This research was supported by the U.S. Department of Energy Contract DE-AC03-76SF00326, PA No. DE-AT03-76ER72018; National Science Foundation Grants MCS-7926009 and ECS-8012974; the Office of Naval Research Contract N00014-75-C-0267; and the U.S. Army Research Office Contract DAAG29-79-C-0110. The work of Nicholas Gould was supported by the Science and Engineering Research Council of Great Britain.     

A proximal-based decomposition method for convex minimization problems

This paper presents a decomposition method for solving convex minimization problems. At each iteration, the algorithm computes two proximal steps in the dual variables and one proximal step in the primal variables. We derive this algorithm from Rockafellar's proximal method of multipliers, which involves an augmented Lagrangian with an additional quadratic proximal term. The algorithm preserves the good features of the proximal method of multipliers, with the additional advantage that it leads to a decoupling of the constraints, and is thus suitable for parallel implementation. We allow for computing approximately the proximal minimization steps and we prove that under mild assumptions on the problem's data, the method is globally convergent and at a linear rate. The method is compared with alternating direction type methods and applied to the particular case of minimizing a convex function over a finite intersection of closed convex sets.Corresponding author. Partially supported by Air Force Office of Scientific Research Grant 91-0008 and National Science Foundation Grant DMS-9201297.     

A variable-penalty alternating directions method for convex optimization

We study a generalized version of the method of alternating directions as applied to the minimization of the sum of two convex functions subject to linear constraints. The method consists of solving consecutively in each iteration two optimization problems which contain in the objective function both Lagrangian and proximal terms. The minimizers determine the new proximal terms and a simple update of the Lagrangian terms follows. We prove a convergence theorem which extends existing results by relaxing the assumption of uniqueness of minimizers. Another novelty is that we allow penalty matrices, and these may vary per iteration. This can be beneficial in applications, since it allows additional tuning of the method to the problem and can lead to faster convergence relative to fixed penalties. As an application, we derive a decomposition scheme for block angular optimization and present computational results on a class of dual block angular problems. This material is based on research supported by the Air Force Office of Scientific Research Grant AFOSR-89-0410 and by NSF Grants CCR-8907671, CDA-9024618 and CCR-9306807.     

A piecewise conservative method for unconstrained convex optimization

Computational Optimization and Applications - We consider a continuous-time optimization method based on a dynamical system, where a massive particle starting at rest moves in the conservative...     

A constraint linearization method for nondifferentiable convex minimization

Summary This paper presents a readily implementable algorithm for solving constrained minimization problems involving (possibly nonsmooth) convex functions. The constraints are handled as in the successive quadratic approximations methods for smooth problems. An exact penalty function is employed for stepsize selection. A scheme for automatic limitation of penalty growth is given. Global convergence of the algorithm is established, as well as finite termination for piecewise linear problems. Numerical experience is reported.Sponsored by Program CPBP 02.15