Robust solutions to uncertain linear complementarity problems

In this paper,we adopt the robust optimization method to consider linear complementarity problems in which the data is not specified exactly or is uncertain,and it is only known to belong to a prescribed uncertainty set.We propose the notion of the p- robust counterpart and the p-robust solution of uncertain linear complementarity problems.We discuss uncertain linear complementarity problems with three different uncertainty sets,respectively,including an unknown-but-bounded uncertainty set,an ellipsoidal uncertainty set and an intersection-of-ellipsoids uncertainty set,and present some sufficient and necessary(or sufficient) conditions which p- robust solutions satisfy.Some special cases are investigated in this paper.     

Two non-probabilistic set-theoretical models to predict the transient vibrations of cross-ply plates with uncertainty

In practical engineering and scientific researches, all engineering analysis and design problems involve uncertainties to various degrees. Dynamic loads acting on a structure are usually with uncertain nature due to the difficulty of predicting the magnitudes of such loads. In this paper, a non-probabilistic and set-theoretical model named interval analysis method is developed to predict the transient vibrations of cross-ply plates with uncertain excitations. The dynamic loads involve deterministic and uncertain components of force function and initial conditions. Uncertainties in these functions are required to be bounded on the L₂ norm and expressed by finite eigenmodes. Analyzed by a numerical example, the width of the upper and lower bounds of the critical buckling loads that calculated by the interval analysis method is sharper than those are obtained by convex models. Moreover, the interval analysis has less computational cost than convex models. Considering specific cases, the effect of various parameters and the level of uncertainty on the response of the cross-ply plates are different.     

Systems framework for fuzzy sets in civil engineering

Civil engineering projects and designs are commonly developed in a systems framework that includes different types of uncertainty. In general, uncertainty can be of the ambiguity or vagueness type. The theory of probability and statistics has been extensively used in civil engineering to deal with the ambiguity type of uncertainty. The theory of fuzzy sets and systems have been used in civil engineering to model the vagueness type of uncertainty in many civil engineering applications. In this paper, the role of fuzzy sets in civil engineering systems is described using several example applications, e.g., quality assessment of wildlife habitat, construction engineering and management, structural reliability, and damage assessment of existing structures.     

Computability in structures representing a Scott set

Continuing work begun in [10], we utilize a notion of forcing for which the generic objects are structures and which allows us to determine whether these “generic” structures compute certain sets and enumerations. The forcing conditions are bounded complexity types which are consistent with a given theory and are elements of a given Scott set. These generic structures will “represent” this given Scott set, in the sense that the structure has a certain weak saturation property with respect to bounded complexity types in the Scott set. For example, if ? is a nonstandard model of PA, then ? represents the Scott set ? = n∈ω | ?⊧“the nth prime divides a” | a∈?. The notion of forcing yields two main results. The first characterizes the sets of natural numbers computable in all models of a given theory representing a given Scott set. We show that the characteristic function of such a set must be enumeration reducible to a complete existential type which is consistent with the given theory and is an element of the given Scott set. The second provides a sufficient condition for the existence of a structure ? such that ? represents a countable jump ideal and ? does not compute an enumeration of a given family of sets ?. This second result is of particular interest when the family of sets which cannot be enumerated is ? = Rep[Th(?)]. Under this additional assumption, the second result generalizes a result on TA [6] and on certain other completions of PA [10]. For example, we show that there also exist models of completions of ZF from which one cannot enumerate the family of sets represented by the theory. Received: 8 October 1997 / Published online: 25 January 2001     

Detection of slightly expressed changes in random environment

Consider a regular parametric family of distributions F(·, θ). The classical change point problem deals with observations corresponding to θ = 0 before a point of change, and θ = μ after that. We substitute the latter constant μ by a set of random variables θ _i,n called a random environment assuming that E[θ _i,n ] = μ _n → 0. The random environment can be independent or obtained by random permutations of a given set. We define the rates of convergence and give the conditions under which the classical parametric change point algorithms apply.     

Dominance-based Rough Set Approach to decision under uncertainty and time preference

We consider a problem of decision under uncertainty with outcomes distributed over time. We propose a rough set model based on a combination of time dominance and stochastic dominance. For the sake of simplicity we consider the case of traditional additive probability distribution over the set of states of the world, however, we show that the model is rich enough to handle non-additive probability distributions, and even qualitative ordinal distributions. The rough set approach gives a representation of decision maker’s time-dependent preferences under uncertainty in terms of “if…, then…” decision rules induced from rough approximations of sets of exemplary decisions.     

Visualizing parametric solution sets

We characterize the boundary ∂Σ ^p of the solution set Σ ^p of a parametric linear system A(p)x=b(p) where the elements of the n×n matrix and the right-hand side vector depend on a number of parameters p varying within interval bounds. The characterization of ∂Σ ^p is by means of pieces of parametric hypersurfaces, the latter represented by their coordinate functions depending on corresponding subsets of n-1 parameters. The presented approach has a direct application for efficient visualization of parametric solution sets by utilizing some plotting functions supported by Mathematica and Maple. Electronic supplementary material The online version of this article (doi: ) contains supplementary material, which is available to authorized users. AMS subject classification (2000) 15A06, 65G99, 65S05, 68U05     

Oligomorphic clones

A permutation group on a countably infinite domain is called oligomorphic if it has finitely many orbits of finitary tuples. We define a clone on a countable domain to be oligomorphic if its set of permutations forms an oligomorphic permutation group. There is a close relationship to ω-categorical structures, i.e., countably infinite structures with a first-order theory that has only one countable model, up to isomorphism. Every locally closed oligomorphic permutation group is the automorphism group of an ω-categorical structure, and conversely, the canonical structure of an oligomorphic permutation group is an ω-categorical structure that contains all first-order definable relations. There is a similar Galois connection between locally closed oligomorphic clones and ω-categorical structures containing all primitive positive definable relations. In this article we generalise some fundamental theorems of universal algebra from clones over a finite domain to oligomorphic clones. First, we define minimal oligomorphic clones, and present equivalent characterisations of minimality, and then generalise Rosenberg’s five types classification to minimal oligomorphic clones. We also present a generalisation of the theorem of Baker and Pixley to oligomorphic clones. Presented by A. Szendrei. Received July 12, 2005; accepted in final form August 29, 2006.     

Strong Formulations of Robust Mixed 0–1 Programming

We introduce strong formulations for robust mixed 0–1 programming with uncertain objective coefficients. We focus on a polytopic uncertainty set described by a ``budget constraint' for allowed uncertainty in the objective coefficients. We show that for a robust 0–1 problem, there is an α–tight linear programming formulation with size polynomial in the size of an α–tight linear programming formulation for the nominal 0–1 problem. We give extensions to robust mixed 0–1 programming and present computational experiments with the proposed formulations.     

Geometric Models for Quasicrystals II. Local Rules Under Isometries

The atomic structures of quasicrystalline materials exhibit long range order under translations. It is believed that such materials have atomic structures which approximately obey local rules restricting the location of nearby atoms. These local constraints are typically invariant under rotations, and it is of interest to establish conditions under which such local rules can nevertheless enforce order under translations in any structure that satisfies them. A set of local rules in is a finite collection of discrete sets {Y _i } containing 0, each of which is contained in the ball of radius ρ around 0 in . A set X satisfies the local rules under isometries if the ρ -neighborhood of each is isometric to an element of . This paper gives sufficient conditions on a set of local rules such that if X satisfies under isometries, then X has a weak long-range order under translations, in the sense that X is a Delone set of finite type. A set X is a Delone set of finite type if it is a Delone set whose interpoint distance set X-X is a discrete closed set. We show for each minimal Delone set of finite type X that there exists a set of local rules such that X satisfies under isometries and all other Y that satisfy under isometries are Delone sets of finite type. A set of perfect local rules (under isometries or under translations, respectively) is a set of local rules such that all structures X that satisfy are in the same local isomorphism class (under isometries or under translations, respectively). If a Delone set of finite type has a set of perfect local rules under translations, then it has a set of perfect local rules under isometries, and conversely. Received February 14, 1997, and in revised form February 14, 1998, February 19, 1998, and March 5, 1998.     

Robust generalized <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript>-suboptimal control

We consider an optimal perturbation damping problem taking into account not only an external perturbation with bounded L ₂-norm and an initial perturbation caused by unknown initial conditions in the system but also unknown bounded parametric perturbations. We synthesize a robust generalized H _∞-suboptimal control minimizing the upper bound, expressed in terms of solutions of linear matrix inequalities, for the perturbation damping level under uncertainty in the closed system.     

Minimax risk over quadratically convex sets

We consider the problem of estimating a vector θ = (θ₁, θ₂,…) ∈ Θ ⊂ l ₂ from observations y _i = θ _i + σ _i x _i , i = 1, 2,…, where the random values x _i are N(0, 1), independent, and identically distributed, the parametric set Θ is compact, orthosymmetric, convex, and quadratically convex. We show that in that case, the minimax risk is not very different from sup?_L( P) \sup {\Re_L}\left( \Pi \right) , where ?_L( P) {\Re_L}\left( \Pi \right) is the minimax linear risk in the same problem with parametric set Π, and sup is taken over all the hyperrectangles Π ⊂ Θ. Donoho, Liu, and McGibbon (1990) have obtained this result for the case of equal σ _i , i = 1, 2,…. Bibliography: 4 titles.     

Tuning of mass dampers for preventing brittle fracture by employing teaching learning based optimization

In design of reinforced concrete structures, a ductile design is needed, because the brittle fracture is sudden. Also, the energy absorbed by the structure is effective for ductile structures during earthquake excitations. In design regulations, especially for vertical supporters such as columns, the shear force and axial force capacity is limited, although the strength of the members is not at the limits. For that reason, the shear forces of structures excited by earthquakes can be reduced by adding a tuned mass damper (TMD), but TMD must be optimized and it must not be heavy. If the mass of the TMD is not optimized, the ductile behaviour constraint about the axial force capacity cannot be provided. The compressive strength of the concrete is a measure for the effective application of a TMD to a reinforced concrete structure. Since the mass is limited by the design constraint, the other design variables of TMD such as period and damping ratio are optimized. Using the formulas for frequency and damping ratio related to a preselected mass may not be sufficient and a precise optimum solution for preventing brittle fracture. Metaheuristic methods can be used by using a specific limit. In this study, Teaching Learning Based Optimization (TLBO) is employed to find optimum TMD parameters. Several earthquake excitations were used in the optimization process. A three storey RC frame structures with different compressive strength of concrete is investigated. Three stopping criteria have been used in the proposed methodology. The first criterion is the reduction of the maxi-mum shear force value for the most critical excitation to the ductile behaviour limit. The second criterion is the reduction of the ratio of maximum first storey displacements of structure with and without TMD. This value is reduced under a user defined value, but the user defined value can be entered as zero. This value can be iteratively increased for minimization. The last criterion is the reduction of the acceleration transfer function of the structure. The proposed method is feasible on finding optimum TMD ensuring ductility conditions. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On artificial geometry errors in the adaptive analysis of eigenvalues and eigenmodes of curved shell structures using the finite element method

The standard procedure to compute design loads for shell structures as also proposed in design rules is based on the computation of the limit load taking into account the modification of the so-called stability loads due to geometrical imperfections. The imperfections are mostly chosen affine to the buckling patterns, which are solutions of the eigenvalue-problem for the geometrically perfect structure. Thus, the eigenvalue-problems for stability points have to be solved very accurately. In the present contribution an adaptive h -refinement procedure is taken for the solution using low order shell elements. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An O(n 2) active set algorithm for the solution of a parametric quadratic program

In this paper, an O(n ²) active set method is presented for minimizing the parametric quadratic function (1/2)x′Dx-a′x + λmax(c - γ x,0) subject to l ⩽ x ⩽ b, for all nonnegative values of the parameter γ. Here, D is a positive diagonal n x n matrix, a and γ are arbitrary N-vectors, c is an arbitrary scalar, l and b are arbitrary n-vectors, such thatl ⩽ b. An extension of this algorithm is presented for minimizing the parametric function (1/2)x′Dx-a ^′ x + λ |γ′x - c| subject to l ⩽ x ⩽ b. It is also shown that these problems arise naturally in a tax programming problem. This revised version was published online in August 2006 with corrections to the Cover Date.     

Generalized convexity and concavity of the optimal value function in nonlinear programming

In this paper we consider generalized convexity and concavity properties of the optimal value functionf ^* for the general parametric optimization problemP(_ε) of the form min _x f(x, ε) s.t.x∈R(ε). Many results on convexity and concavity characterizations off ^* were presented by the authors in a previous paper. Such properties off ^* and the solution set mapS ^* form an important part of the theoretical basis for sensitivity, stability and parametric analysis in mathematical optimization. We give sufficient conditions for several types of generalized convexity and concavity off ^*, in terms of respective generalized convexity and concavity assumptions onf and convexity and concavity assumptions on the feasible region point-to-set mapR. Specializations of these results to the parametric inequality-equality constrained nonlinear programming problem are provided. Research supported by Grant ECS-8619859, National Science Foundation and Contract N00014-86-K-0052, Office of Naval Research.     

The role of fuzzy sets in civil engineering

Civil Engineering is an art, the practise of which requires the use of scientific knowledge as a basic tool. The engineer has to use considerable judgement in setting up and interpreting his scientific calculations and in making decisions based upon incomplete information. Because a very high safety level is required in civil engineering structures the uncertainty associated with the application of scientific calculations is very crudely and conservatively estimated using traditional methods. Current reliability theory enables a discussion of random parameter uncertainty but in civil engineering, system uncertainty and the possibility of human error is extremely important and must be included in any reliability calculations.The potential role of fuzzy sets in analysing system and human uncertainty is discussed in the paper and two numerical examples are given of the calculation of system and random uncertainty using probability measures of fuzzy sets.     

A review of major paradigms and models for the design of civil engineering systems

In this paper, the author presents the five classical paradigms of the process of design in civil engineering and identifies a new emerging paradigm: the interactive multi-attribute learning paradigm. This paradigm is studied in terms of actors, structures and OR instruments which can help to fulfil its application to modern design of civil engineering systems.     

<Emphasis Type="Italic">r</Emphasis>-Tuple almost product structures

A generalization of an almost product structure and an almost complex structure on smooth manifolds is constructed. The set of tensor differential invariants of type (2, 1) and the set of differential 2-forms for such structures are constructed. We show how these tensor invariants can be used to solve the classification problem for Monge–Ampère equations and Jacobi equations.