Showing papers on "Solution set published in 2004"
01 Jan 2004
TL;DR: In this paper, the authors define the following concepts: a function from a set X into a set Y, denoted by Function of X,Y ", the set of all functions from X into Y, denoted as Funcs(X,Y ), and the permutation of a set (mode Permutation of X, where X is a set).
Abstract: The article is a continuation of [1]. We define the following concepts: a function from a set X into a set Y , denoted by “Function of X ,Y ”, the set of all functions from a set X into a set Y , denoted by Funcs(X ,Y ), and the permutation of a set (mode Permutation of X , where X is a set). Theorems and schemes included in the article are reformulations of the theorems of [1] in the new terminology. Also some basic facts about functions of two variables are proved.
352 citations
19 Apr 2004
TL;DR: Different fuzzy-based definitions of optimality and dominated solutions, being nonpreference based, are introduced and tested on analytical test cases, in order to show their validity and nearness to human decision making.
Abstract: When dealing with many-objectives optimization problems, the concepts of Pareto-optimality and Pareto-dominance are often inefficient in modeling and simulating human decision making. This leads to an unpractical size for the set of Pareto-optimal (PO) solutions, and an additional selection criteria among solutions is usually arbitrarily considered. In the paper, different fuzzy-based definitions of optimality and dominated solutions, being nonpreference based, are introduced and tested on analytical test cases, in order to show their validity and nearness to human decision making. Based on this definitions, different subsets of PO solution set can be computed using simple and clear information provided by the decision maker and using a parameter value ranging from zero to one. When the value of the above parameter is zero, the introduced definitions coincide with classical Pareto-optimality and dominance. When the parameter value is increased, different subset of PO solutions can be obtained corresponding to higher degrees of optimality.
316 citations
TL;DR: In this article, the authors established sufficient conditions for the solution set of parametric multivalued vector quasiequilibrium problems to be semicontinuous, and investigated both the weak and strong solutions.
134 citations
TL;DR: Semidefinite relaxation techniques for computing minimal size ellipsoids that bound the solution set of a system of uncertain linear equations and can be directly applied to various state-bounding problems for dynamical systems subject to set-valued noise and model uncertainty.
94 citations
TL;DR: It is proved in this paper that the proposed smoothing Newton algorithm, which is a modified version of the Qi-Sun-Zhou algorithm, has the following convergence properties: it is well-defined and any accumulation point of the iteration sequence is a solution of the P0–LCP.
Abstract: Given *** equation here ***, the linear complementarity problem (LCP) is to find *** equation here *** such that (x, s)≥ 0,s=Mx+q,xTs=0. By using the Chen-Harker-Kanzow-Smale (CHKS) smoothing function, the LCP is reformulated as a system of parameterized smooth-nonsmooth equations. As a result, a smoothing Newton algorithm, which is a modified version of the Qi-Sun-Zhou algorithm [Mathematical Programming, Vol. 87, 2000, pp. 1–35], is proposed to solve the LCP with M being assumed to be a P0-matrix (P0–LCP). The proposed algorithm needs only to solve one system of linear equations and to do one line search at each iteration. It is proved in this paper that the proposed algorithm has the following convergence properties: (i) it is well-defined and any accumulation point of the iteration sequence is a solution of the P0–LCP; (ii) it generates a bounded sequence if the P0–LCP has a nonempty and bounded solution set; (iii) if an accumulation point of the iteration sequence satisfies a nonsingularity condition, which implies the P0–LCP has a unique solution, then the whole iteration sequence converges to this accumulation point sub-quadratically with a Q-rate 2–t, where t∈(0,1) is a parameter; and (iv) if M is positive semidefinite and an accumulation point of the iteration sequence satisfies a strict complementarity condition, then the whole sequence converges to the accumulation point quadratically.
80 citations
TL;DR: In this paper, the authors considered the GMVI (f′,K) with radially l.c.s. function f and showed that the solution set of the problem is a convex and radially closed subset of the k-subset of the K-subspace.
Abstract: Let E be a linear space, let K \(\subseteq\) E and f:K→ℝ . We formulate in terms of the lower Dini directional derivative problem GMVI (f′,K), which can be considered as a generalization of MVI (f′,K), the Minty variational inequality of differential type. We investigate, in the case of K star-shaped (SS), the existence of a solution x* of GMVI (f′K) and the property of f to increase-along-rays starting at x*, f∈IAR (K,x*). We prove that the GMVI (f′,K) with radially l.s.c. function f has a solution x*∈ ker K if and only if f∈IAR (K,x*). Further, we prove that the solution set of the GMVI (f′,K) is a convex and radially closed subset of ker K. We show also that, if the GMVI (f′,K) has a solution x*∈K, then x* is a global minimizer of the problem min f(x), x∈K. Moreover, we observe that the set of the global minimizers of the related optimization problem, its kernel, and the solution set of the variational inequality can be different. Finally, we prove that, in the case of a quasiconvex function f, these sets coincide.
64 citations
TL;DR: This work constructs two fixed-point iteration algorithms that solve convex subproblems and that are guaranteed, for sufficiently small friction coefficients, to retrieve the unique velocity solution of the nonconvex linear complementarity problem whenever the frictionless configuration can be disassembled.
Abstract: Acceleration–force setups for multi-rigid-body dynamics are known to be inconsistent for some configurations and sufficiently large friction coefficients (a Painleve paradox). This difficulty is circumvented by time-stepping methods using impulse-velocity approaches, which solve complementarity problems with possibly nonconvex solution sets. We show that very simple configurations involving two bodies may have a nonconvex solution set for any nonzero value of the friction coefficient. We construct two fixed-point iteration algorithms that solve convex subproblems and that are guaranteed, for sufficiently small friction coefficients, to retrieve, at a linear convergence rate, the unique velocity solution of the nonconvex linear complementarity problem whenever the frictionless configuration can be disassembled. In addition, we show that one step of one of the iterative algorithms provides an excellent approximation to the velocity solution of the original, possibly nonconvex, problem if for all contacts we have that either the friction coefficient is small or the slip velocity is small.
62 citations
TL;DR: This paper presents an application of a linear complementarity problem where M is a P-matrix but, in general, is neither an H-Matrix nor a positive definite matrix, and extends the idea such that the calculations can be done by a computer with rigorous error control.
Abstract: In this paper, we present an application of a linear complementarity problem where M is a P-matrix but, in general, is neither an H-matrix nor a positive definite matrix. This application occurs originally in [J. Rohn, Linear Algebra Appl., 126 (1989), pp. 39--78], which is less known to the LCP community. Its focus is in computing the exact interval enclosures of the components of the solution set of an interval linear system. We extend the idea such that the calculations can be done by a computer with rigorous error control.
62 citations
TL;DR: The aim of this paper is to argue in favour of parameterized quadratic objective functions, in contrast to the standard weighting approach in which parameterized linear objective functions are used.
Abstract: In multicriteria optimization, several objective functions have to be minimized simultaneously. For this kind of problem, approximations to the whole solution set are of particular importance to decision makers. Usually, approximating this set involves solving a family of parameterized optimization problems. It is the aim of this paper to argue in favour of parameterized quadratic objective functions, in contrast to the standard weighting approach in which parameterized linear objective functions are used. These arguments will rest on the favourable numerical properties of these quadratic scalarizations, which will be investigated in detail. Moreover, it will be shown which parameter sets can be used to recover all solutions of an original multiobjective problem where the ordering in the image space is induced by an arbitrary convex cone.
59 citations
Patent•
20 Feb 2004
TL;DR: In this article, a method for multi-objective portfolio optimization for use in investment decisions based on competing objectives and a plurality of constraints constituting a portfolio problem is presented. But this method is not suitable for the problem of multi-dimensional portfolio optimization.
Abstract: The systems and methods of the invention are directed to portfolio optimization and related techniques. For example, the invention provides a method for multi-objective portfolio optimization for use in investment decisions based on competing objectives and a plurality of constraints constituting a portfolio problem, the method sequentially comprising: generating a non-dominated solution set in a space; applying a first set of user-specified constraints to reduce the solutions in the non-dominated solution set to a solution subset; and executing a series of local tradeoffs on the solution subset to result in a resulting solution subset, the local tradeoffs being performed in a lower dimension performance space as compared to the space, and the solution subset being used in investment decisioning.
52 citations
TL;DR: A number of techniques to improve the execution time of interprocedural pointer analysis in the context of C programs are presented and a novel online algorithm for doing this is presented.
Abstract: This paper presents and evaluates a number of techniques to improve the execution time of interprocedural pointer analysis in the context of C programs. The analysis is formulated as a graph of set constraints and solved using a worklist algorithm. Indirections lead to new constraints being added during this procedure. The solution process can be simplified by identifying cycles, and we present a novel online algorithm for doing this. We also present a difference propagation scheme which avoids redundant work by tracking changes to each solution set. The effectiveness of these and other methods are shown in an experimental study over 12 common ‘C’ programs ranging between 1000 to 150,000 lines of code.
TL;DR: The paper deals with the problem of determining an outer interval solution (interval enclosure of the solution set) of linear systems whose elements are affine functions of interval parameters and an iterative method for finding such a solution is suggested.
Abstract: The paper deals with the problem of determining an outer interval solution (interval enclosure of the solution set) of linear systems whose elements are affine functions of interval parameters. An iterative method for finding such a solution is suggested. A numerical example illustrating the new method is solved.
TL;DR: The results suggest that the GR algorithm provides an efficient way to identify subsets of preferred Pareto optima from larger sets in multi-objective optimization.
Abstract: Algorithms for multi-objective optimization problems are designed to generate a single Pareto optimum (non-dominated solution) or a set of Pareto optima that reflect the preferences of the decision-maker. If a set of Pareto optima are generated, then it is useful for the decision-maker to be able to obtain a small set of preferred Pareto optima using an unbiased technique of filtering solutions. This suggests the need for an efficient selection procedure to identify such a preferred subset that reflects the preferences of the decision-maker with respect to the objective functions. Selection procedures typically use a value function or a scalarizing function to express preferences among objective functions. This paper introduces and analyzes the Greedy Reduction (GR) algorithm for obtaining subsets of Pareto optima from large solution sets in multi-objective optimization. Selection of these subsets is based on maximizing a scalarizing function of the vector of percentile ordinal rankings of the Pareto optima within the larger set. A proof of optimality of the GR algorithm that relies on the non-dominated property of the vector of percentile ordinal rankings is provided. The GR algorithm executes in linear time in the worst case. The GR algorithm is illustrated on sets of Pareto optima obtained from five interactive methods for multi-objective optimization and three non-linear multi-objective test problems. These results suggest that the GR algorithm provides an efficient way to identify subsets of preferred Pareto optima from larger sets.
01 Jan 2004
TL;DR: In this contribution global observability of nonlinear systems is investigated and a criterion which allows to decide if two initial states of a system can be distinguished by the output of the system is derived.
Abstract: In this contribution global observability of nonlinear systems is investigated. The main idea is to derive a criterion which allows to decide if two initial states of a system can be distinguished by the output of the system. This well known criterion is equivalent to an infinite set of nonlinear equations. If the system is globally observable this set of equations has a very special solution set which can be characterized algebraically. Based on results from commutative algebra some new algorithmic methods to describe the solution set are given. This allows a global observability analysis for polynomial systems and for some specific classes of nonlinear systems. The new method is applied to an example from the literature.
Patent•
27 Aug 2004
TL;DR: In this article, the authors present methods, systems, and processor program products that include executing an optimization scheme to obtain a first solution set, presenting the solution set to at least two users, receiving rankings of the first solution sets from the users, aggregating the rankings, and, generating a second solution set based on the aggregated rankings.
Abstract: Disclosed are methods, systems, and processor program products that include executing an optimization scheme to obtain a first solution set, presenting the first solution set to at least two users, receiving rankings of the first solution set from the at least two users, aggregating the rankings, and, generating a second solution set based on the aggregated rankings. The optimization scheme can include a genetic algorithm. In embodiments, at least a part of the first solution set can be presented to the users based on the parts of the solution set associated with the user (e.g., user's knowledge).
TL;DR: In this article, the authors derived necessary and sufficient conditions for pseudomonotone and quasimonotone variational inequalities to have a nonempty, possibly unbounded solution set.
Abstract: A coercivity condition is usually assumed in variational inequalities over noncompact domains to guarantee the existence of a solution. We derive minimal, i.e., necessary coercivity conditions for pseudomonotone and quasimonotone variational inequalities to have a nonempty, possibly unbounded solution set. Similarly, a minimal coercivity condition is derived for quasimonotone variational inequalities to have a nonempty, bounded solution set, thereby complementing recent studies for the pseudomonotone case. Finally, for quasimonotone complementarity problems, previous existence results involving so-called exceptional families of elements are strengthened by considerably weakening assumptions in the literature.
TL;DR: Numerical examples illustrating the superiority of L(x,y)=Ax+By+b over L( x)=Ax-By-b have been solved for the case where the problem is the global solution of a system of nonlinear equations (n=m).
Abstract: Let f(x) denote a system of n nonlinear functions in m variables, m≥n Recently, a linearization of f(x) in a box x has been suggested in the form L(x)=Ax+b where A is a real n×m matrix and b is an interval n-dimensional vector Here, an improved linearization L(x,y)=Ax+By+b, x∈x, y∈y is proposed where y is a p-dimensional vector belonging to the interval vector y while A and B are real matrices of appropriate dimensions and b is a real vector The new linearization can be employed in solving various nonlinear problems: global solution of nonlinear systems, bounding the solution set of underdetermined systems of equations or systems of equalities and inequalities, global optimization Numerical examples illustrating the superiority of L(x,y)=Ax+By+b over L(x)=Ax+b have been solved for the case where the problem is the global solution of a system of nonlinear equations (n=m)
TL;DR: In this article, a strategy for using linear programming (LP) to bound the solution set of the linear interval equation system that must be solved in the context of the interval-Newton method for deterministic linear systems is described.
Abstract: A strategy is described for using linear programming (LP) to bound the solution set of the linear interval equation system that must be solved in the context of the interval-Newton method for deter...
12 Feb 2004
TL;DR: A new efficient method for approximating the solution set of a convex quadratic multiobjective programming problem is proposed, based on a warm-start interior point algorithm for which complexity results are derived, thereby extending previous results by Yildirim & Wright.
Abstract: In multicriteria optimization, several objective functions have to be minimized simultaneously. For this kind of problem, no single solution can adequately represent the whole set of optimal points. We propose a new efficient method for approximating the solution set of a convex quadratic multiobjective programming problem. The method is based on a warm-start interior point algorithm for which we derive complexity results, thereby extending previous results by Yildirim & Wright. Numerical results on bicriteria problems from power plant optimization and portfolio optimization show that the method is an order of magnitude faster than standard methods applied to the problems considered.
TL;DR: In this paper, the authors presented the new concept of exceptional family of elements for the variational inequality problem with a continuous function over a general unbounded closed convex set. And they established a characterization theorem that can be used to derive several new existence and compactness conditions on the solution set.
Abstract: This paper presents the new concept of exceptional family of elements for the variational inequality problem with a continuous function over a general unbounded closed convex set. We establish a characterization theorem that can be used to derive several new existence and compactness conditions on the solution set. Our findings generalize well-known results for various types of variational inequality problems. For a pseudomonotone variational inequality problem, our new existence conditions are both sufficient and necessary.
27 Sep 2004
TL;DR: In this paper, it is shown that it is possible to represent the domains of finite set variables using reduced ordered binary decision diagrams (ROBDDs) and furthermore to build efficient domain propagators for set constraints using ROBDDs.
Abstract: Propagation based solvers typically represent variables by a current domain of possible values that may be part of a solution. Finite set variables have been considered impractical to represent as a domain of possible values since, for example, a set variable ranging over subsets of {1, ..., N} has 2N possible values. Hence finite set variables are presently represented by a lower bound set and upper bound set, illustrating all values definitely in (and by negation) all values definitely out. Propagators for finite set variables implement set bounds propagation where these sets are further constrained. In this paper we show that it is possible to represent the domains of finite set variables using reduced ordered binary decision diagrams (ROBDDs) and furthermore we can build efficient domain propagators for set constraints using ROBDDs. We show that set domain propagation is not only feasible, but can solve some problems significantly faster than using set bounds propagation because of the stronger propagation.
TL;DR: In this article, the authors define weak sharp solutions to vector optimization problems and the growth condition for vector-valued functions, and prove upper Holder continuity and Holder calmness of the solution set-valued mapping to parametric vector optimisation problems.
Abstract: We define weak sharp solutions to vector optimization problems and the growth condition for vector-valued functions. When applied to scalar-valued functions, weak sharp solutions reduce to weak sharp minima, and the growth condition reduces to the growth condition used in proving Holder calmness of the solution set to parametric scalar optimization problems. By using these concepts We prove upper Holder continuity and Holder calmness of the solution set-valued mapping to parametric vector optimization problems.
TL;DR: Some verifiable conditions that are applicable to sets with finite number of elements, to corroborate or falsify the hypothesis of the elements of that set being samples of the Pareto set are proposed.
Abstract: Multiobjective design problems give rise to a well-defined object: the Pareto-set. This paper proposes some verifiable conditions that are applicable to sets with finite number of elements, to corroborate or falsify the hypothesis of the elements of that set being samples of the Pareto set. These conditions lead to several generic criteria that can be employed in the evaluation of algorithms as multiobjective optimization mechanisms. A conceptual multiobjective genetic algorithm is proposed, exploiting the group properties of the intermediate Pareto-set estimates to generate a consistent final estimate. The methodology is applied to the case of a mixed H2/H∞ control design. Recent dedicated multiobjective algorithms are evaluated under the proposed methodology, and it is shown that they can generate sub-optimal or non-consistent solution sets. It is shown that the proposed synthesis methodology can lead to both enhanced objectives and enhanced consistency in the Pareto-set estimate.
TL;DR: The behavior and sensitivity analysis of a solution set for a new class of generalized nonlinear implicit quasi-variational inclusions is studied by using a resolvent operator technique of maximal monotone mappings and the property of a fixed-point set of set-valued contractive mappings.
TL;DR: The method obtains an estimate of the Pareto frontier with maximum solution diversity using a quality index, referred to as entropy index, and improves overall quality of solutions by explicitly optimizing the entropy index at every system-level iteration.
Abstract: This article presents a new method for multiobjective multidisciplinary design optimization. The method obtains an estimate of the Pareto frontier with maximum solution diversity using a quality index, referred to as entropy index. Unlike previous methods that maintain diversity in the solution set heuristically, our method improves overall quality of solutions by explicitly optimizing the entropy index at every system-level iteration, and then using this information to bias the search process toward obtaining a solution set with maximum diversity. Our method utilizes a multiobjective genetic algorithm as an optimizer in each subproblem of a multidisciplinary optimization problem. To demonstrate the proposed approach, we applied our method to a mechanical design problem of a speed reducer and the results are compared with those obtained by a few other multiobjective optimization methods. A minimal set of quality indexes is used to compare the diversity and optimality of the obtained solution sets...
01 Jan 2004
TL;DR: In this paper, a necessary condition for any minimal solution in terms of the maximum solution is provided, where each nonzero component of a minimal solution takes the value of corresponding component of the maximal solution.
Abstract: It is well known that the solution set of fuzzy relational equations with max-product composition can be determined by the maximum solution and a finite number of minimal solutions There exists an analytic expression for the maximum solution and it can be yielded easily, but finding the complete set of minimal solutions is not trivial In this paper we first provide a necessary condition for any minimal solution in terms of the maximum solution Precisely, each nonzero component of any minimal solution takes the value of corresponding component of the maximum solution We then propose rules to reduce the problem so that the solution can easily be found A numerical example is provided to illustrate our procedure
TL;DR: A simple approach is proposed for some particular models of interval uncertainty that gives an optimal interval solution without linear programming and is tractable for moderate-size problems.
TL;DR: This work considers equilibrium configurations of inextensible, unshearable, isotropic, uniform and naturally straight and prismatic rods when subject to end loads and clamped boundary conditions and calls this set the solution manifold, which is the set of all parameter values that yield equilibrium configurations that fulfill clamped boundaries conditions.
Abstract: We consider equilibrium configurations of inextensible, unshearable, isotropic, uniform and naturally straight and prismatic rods when subject to end loads and clamped boundary conditions. In a first paper [Neukirch & Henderson, 2002], we discussed symmetry properties of the equilibrium configurations of the center line of the rod. Here, we are interested in the set of all parameter values that yield equilibrium configurations that fulfill clamped boundary conditions. We call this set the solution manifold and we compute it using a recently introduced continuation algorithm. We then describe the topology of this manifold and how it comprises different interconnected layers. We show that the border set of the different layers is the well-known solution set of buckled rings.
TL;DR: An analogue of the classical Bogolyubov theorem, with a nonconvex constraint, is proved, based on a relaxation argument, as in the Filippov--Wazewski theorem, about a solution set of a Cauchy problem for a differential inclusion satisfying a Lipschitz condition.
Abstract: We prove an analogue of the classical Bogolyubov theorem, with a nonconvex constraint. In the case we consider, the constraint is the solution set of a Cauchy problem for a differential inclusion with a nonconvex right-hand side satisfying a Lipschitz condition. Our approach is based on a relaxation argument, as in the Filippov--Wazewski theorem.
01 Jan 2004
TL;DR: In this paper, the existence of non-unimmodular extended eigenvalues leads to invariant or hyper-invariant subspaces for a bounded operator A on a complex Hilbert space H, and the set EE(A, λ) is defined to be the set of those complex numbers λ for which there is an operator T = 0 satisfying AT = λTA.
Abstract: An extended eigenvalue for an operator A is a scalar λ for which the operator equation AX = λXA has a nonzero solution. Several scenarios are investigated where the existence of non-unimodular extended eigenvalues leads to invariant or hyperinvariant subspaces. For a bounded operator A on a complex Hilbert space H, the set EE(A) of extended eigenvalues for A is defined to be the set of those complex numbers λ for which there is an operator T = 0 satisfying AT = λTA. T is then referred to as a λ eigen-operator for A. The eigenvalue terminology, although not perfectly accurate, seems useful on two levels. The first was described in [2]; briefly, if A has dense range, then the equation AX = φ(X)A; φ(X) ∈ L(H) has a unital algebra as its solution set, and φ is a unital homomorphism. Our extended eigenvalues are precisely the eigenvalues for φ. The second point of view is that one can easily show that for an operator on a finite dimensional space, the set of extended eigenvalues for that operator is the set of ratios of eigenvalues, with the obvious restriction on the use of 0. This is shown explicitly in [3]. In other works this concept of extended eigenvalue has appeared as α commuting or λ commuting, but we choose to use a term which is parameter free. For A ∈ L(H) (the set of bounded operators on H), a (closed, linear) subspace of H is a nontrivial invariant subspace (n.i.s.) for A if it is neither H nor {0} and is invariant under A. This space is hyperinvariant for A if it is invariant for every operator in (A)′, the commutant of A. More generally, a subspace is defined to be invariant for a set of operators if it is invariant for each member of that set. Extended eigenvalues and invariant subspaces. For a given λ ∈ EE(A) we define E = E(A, λ) as the set of all λ eigen-operators for A. This is a (weakly) closed linear space of operators, and E(A, 1) is (A)′, the commutant of A; that is, the set of all operators commuting with A. Direct multiplication leads to the next result: Received June 15, 2003. Mathematics Subject Classification. 47A15, 47A62.