Showing papers on "Solution set published in 2008"
TL;DR: In this article, the authors considered the minimization of transmit power in Gaussian parallel interference channels, subject to a rate constraint for each user, and derived sufficient conditions that guarantee the existence and nonemptiness of the solution set.
Abstract: This paper considers the minimization of transmit power in Gaussian parallel interference channels, subject to a rate constraint for each user. To derive decentralized solutions that do not require any cooperation among the users, we formulate this power control problem as a (generalized) Nash equilibrium (NE) game. We obtain sufficient conditions that guarantee the existence and nonemptiness of the solution set to our problem. Then, to compute the solutions of the game, we propose two distributed algorithms based on the single user water-filling solution: The sequential and the simultaneous iterative water-filling algorithms, wherein the users update their own strategies sequentially and simultaneously, respectively. We derive a unified set of sufficient conditions that guarantee the uniqueness of the solution and global convergence of both algorithms. Our results are applicable to all practical distributed multipoint-to-multipoint interference systems, either wired or wireless, where a quality of service in (QoS) terms of information rate must be guaranteed for each link.
214 citations
TL;DR: Some of the assumptions of the problem are discussed, under which the introduced conditions are sufficient and/or necessary, and the effect of these assumptions on the connection between the solution sets of the equilibrium problem and of a related convex feasibility problem is analyzed.
Abstract: The main purpose of this paper is the study of sufficient and/or necessary conditions for existence of solutions of equilibrium problems. We discuss some of the assumptions of the problem, under which the introduced conditions are sufficient and/or necessary, and also analyze the effect of these assumptions on the connection between the solution sets of the equilibrium problem and of a related convex feasibility problem.
125 citations
TL;DR: A new error bound for the monotone LCP is given and used to show that solutions of the ERM formulation are robust in the sense that they may have a minimum sensitivity with respect to random parameter variations in SLCP.
Abstract: We consider the stochastic linear complementarity problem (SLCP) involving a random matrix whose expectation matrix is positive semi-definite. We show that the expected residual minimization (ERM) formulation of this problem has a nonempty and bounded solution set if the expected value (EV) formulation, which reduces to the LCP with the positive semi-definite expectation matrix, has a nonempty and bounded solution set. We give a new error bound for the monotone LCP and use it to show that solutions of the ERM formulation are robust in the sense that they may have a minimum sensitivity with respect to random parameter variations in SLCP. Numerical examples including a stochastic traffic equilibrium problem are given to illustrate the characteristics of the solutions.
122 citations
TL;DR: In this article, the problem of minimizing a linear objective function subject to a system of sup-T equations can be reduced into a 0 − 1 integer programming problem in polynomial time.
Abstract: This paper provides a thorough investigation on the resolution of a finite system of fuzzy relational equations with sup-T composition, where T is a continuous triangular norm. When such a system is consistent, although we know that the solution set can be characterized by a maximum solution and finitely many minimal solutions, it is still a challenging task to find all minimal solutions in an efficient manner. Using the representation theorem of continuous triangular norms, we show that the systems of sup-T equations can be divided into two categories depending on the involved triangular norm. When the triangular norm is Archimedean, the minimal solutions correspond one-to-one to the irredundant coverings of a set covering problem. When it is non-Archimedean, they only correspond to a subset of constrained irredundant coverings of a set covering problem. We then show that the problem of minimizing a linear objective function subject to a system of sup-T equations can be reduced into a 0---1 integer programming problem in polynomial time. This work generalizes most, if not all, known results and provides a unified framework to deal with the problem of resolution and optimization of a system of sup-T equations. Further generalizations and related issues are also included for discussion.
107 citations
TL;DR: In this article, the authors studied the set of solutions of random k-satisfiability formulas through the cavity method and showed that for an interval of the clause-to-variables ratio, this decomposes into an exponential number of pure states (clusters).
Abstract: We study the set of solutions of random k-satisfiability formulas through the cavity method. It is known that, for an interval of the clause-to-variables ratio, this decomposes into an exponential number of pure states (clusters). We refine substantially this picture by: (i) determining the precise location of the clustering transition; (ii) uncovering a second 'condensation' phase transition in the structure of the solution set for k≥4. These results both follow from computing the large deviation rate of the internal entropy of pure states. From a technical point of view our main contributions are a simplified version of the cavity formalism for special values of the Parisi replica symmetry breaking parameter m (in particular for m = 1 via a correspondence with the tree reconstruction problem) and new large-k expansions.
90 citations
TL;DR: In this article, the authors provide sufficient conditions for the continuity of the solution set mapping in parametric weak monotone vector equilibrium problems in topological vector spaces, and obtain some stability results for these problems.
Abstract: In this paper, we obtain some stability results for parametric weak vector equilibrium problems in topological vector spaces. We provide sufficient conditions for the continuity of the solution set mapping in parametric weak monotone vector equilibrium problems.
89 citations
TL;DR: A simplified version of the cavity formalism for special values of the Parisi replica symmetry breaking parameter m (in particular for m = 1 via a correspondence with the tree reconstruction problem) and new large-k expansions are made.
Abstract: We study the set of solutions of random k-satisfiability formulae through the cavity method. It is known that, for an interval of the clause-to-variables ratio, this decomposes into an exponential number of pure states (clusters). We refine substantially this picture by: (i) determining the precise location of the clustering transition; (ii) uncovering a second `condensation' phase transition in the structure of the solution set for k larger or equal than 4. These results both follow from computing the large deviation rate of the internal entropy of pure states. From a technical point of view our main contributions are a simplified version of the cavity formalism for special values of the Parisi replica symmetry breaking parameter m (in particular for m=1 via a correspondence with the tree reconstruction problem) and new large-k expansions.
89 citations
TL;DR: It is proved that for a given nonsmooth convex optimization problem and sufficiently large penalty parameters, any trajectory of the neural network can reach the feasible region in finite time and stays there thereafter.
Abstract: This paper develops a neural network for solving the general nonsmooth convex optimization problems. The proposed neural network is modeled by a differential inclusion. Compared with the existing neural networks for solving nonsmooth convex optimization problems, this neural network has a wider domain for implementation. Under a suitable assumption on the constraint set, it is proved that for a given nonsmooth convex optimization problem and sufficiently large penalty parameters, any trajectory of the neural network can reach the feasible region in finite time and stays there thereafter. Moreover, we can prove that the trajectory of the neural network constructed by a differential inclusion and with arbitrarily given initial value, converges to the set consisting of the equilibrium points of the neural network, whose elements are all the optimal solutions of the primal constrained optimization problem. In particular, we give the condition that the equilibrium point set of the neural network coincides with the optimal solution set of the primal constrained optimization problem and the condition ensuring convergence to the optimal solution set in finite time. Furthermore, illustrative examples show the correctness of the results in this paper, and the good performance of the proposed neural network.
88 citations
TL;DR: In this paper, a general fixed point iteration for computing a point in some nonempty closed and convex solution set included in the common fixed point set of a sequence of mappings on a real Hilbert space is proposed.
71 citations
TL;DR: In this paper, the authors investigate topological properties and stability of solution sets in parametric variational relation problems and give a unifying way to treat these questions in the theory of variational inequalities, variational inclusions and equilibrium problems.
Abstract: The purpose of this paper is to investigate topological properties and stability of solution sets in parametric variational relation problems The results of the paper give a unifying way to treat these questions in the theory of variational inequalities, variational inclusions and equilibrium problems
68 citations
TL;DR: All maximal solutions are characterized and it is proved that under certain conditions each solution of the system is less than or equal to a respective maximal one.
TL;DR: This work proposes and analyzes two different archiving strategies which lead to a different limit behavior of the algorithms, yielding bounds on the obtained approximation quality as well as on the cardinality of the resulting Pareto set approximation.
Abstract: In this work we investigate the convergence of stochastic search algorithms toward the Pareto set of continuous multi-objective optimization problems. The focus is on obtaining a finite approximation that should capture the entire solution set in a suitable sense, which will be defined using the concept of ?-dominance. Under mild assumptions about the process to generate new candidate solutions, the limit approximation set will be determined entirely by the archiving strategy. We propose and analyse two different archiving strategies which lead to a different limit behavior of the algorithms, yielding bounds on the obtained approximation quality as well as on the cardinality of the resulting Pareto set approximation.
TL;DR: It is shown that a solution set comprises both attainable and unattainable solutions, which are closely related to minimal solutions to the equations.
TL;DR: An existence theorem of solutions for the generalized strong vector quasi-equilibrium problem is established by using Kakutani-Fan-Glicksberg fixed point theorem and the closedness of the strong solution set is discussed.
12 Jul 2008
TL;DR: This paper explains elementary landscapes in terms of the expected value of solution components which are transformed in the process of moving from an incumbent solution to a neighboring solution.
Abstract: The landscape formalism unites a finite candidate solution set to a neighborhood topology and an objective function. This construct can be used to model the behavior of local search on combinatorial optimization problems. A landscape is elementary when it possesses a unique property that results in a relative smoothness and decomposability to its structure. In this paper we explain elementary landscapes in terms of the expected value of solution components which are transformed in the process of moving from an incumbent solution to a neighboring solution. We introduce new results about the properties of elementary landscapes and discuss the practical implications for search algorithms.
TL;DR: In this article, the generalized reflexive solution of matrix equations (AX=B, XC=D ) is considered and the necessary and sufficient conditions for the solvability and the general expression of the solution are obtained.
TL;DR: Sufficient conditions for the solution sets of parametric multivalued symmetric vector quasiequilibrium problems to have semicontinuity-related properties are established.
Abstract: We introduce some definitions related to semicontinuity of multivalued mappings and discuss various kinds of semicontinuity-related properties. Sufficient conditions for the solution sets of parametric multivalued symmetric vector quasiequilibrium problems to have these properties are established. Comparisons of the solution sets of our two problems are also provided. As an example of applications of our main results, the mentioned semicontinuity-related properties of the solution sets to a lower and upper bounded quasiequilibrium problem are obtained as consequences.
TL;DR: Lai et al. as discussed by the authors proposed a fuzzy Pareto optimal set to solve the multicriteria decision making problem in water resource, which can capture the uncertainty associated with the assignment of weights and provide the decision makers with a wider range of solutions to select from.
Abstract: A multiciteria decision making problem in water resource is addressed through the generation of fuzzy Pareto optimal set. Methodology is using positive and negative ideal solutions (Lai, Y.-J., Liu, T.-Y., Hwang, C.L. (1994). TOPSIS for MODM. European Journal of Operational Research 76, 486–500) and a set of weights assigned to the objective functions in fuzzy triangular form. Solution of the problem is obtained by transforming each objective function into a set of three objective functions. A planning problem of multicriteria waste water treatment from the literature is used as an illustrative example to demonstrate the utility of the proposed methodology. The obtained fuzzy Pareto solution set has been compared with the deterministic solution set. It is shown that the proposed approach can: (a) capture the uncertainty associated with the assignment of weights; and (b) provide the decision makers with a wider range of solutions to select from.
TL;DR: An optimal approximation between a given matrix X@?@?C^m^x^n and the affine subspace S"X is discussed, an explicit formula for the unique optimal approximation solution is presented, and a numerical example is provided.
Abstract: Let R@?C^m^x^m and S@?C^n^x^n be nontrivial unitary involutions, i.e., R^H=R=R^-^1 I"m and S^H=S=S^-^1 I"n. We say that G@?C^m^x^n is a generalized reflexive matrix if RGS=G. The set of all mxn generalized reflexive matrices is denoted by GRC^m^x^n. In this paper, a sufficient and necessary condition for the matrix equation AXB=D, where A@?C^p^x^m,B@?C^n^x^q and D@?C^p^x^q, to have a solution X@?GRC^m^x^n is established, and if it exists, a representation of the solution set S"X is given. An optimal approximation between a given matrix X@?@?C^m^x^n and the affine subspace S"X is discussed, an explicit formula for the unique optimal approximation solution is presented, and a numerical example is provided.
TL;DR: In this article, the existence of mild solutions for the impulsive Cauchy problem on non-compact domains has been studied and a theorem on the compactness of the set of all mild solutions is given.
Abstract: This paper deals with an impulsive Cauchy problem governed by the semilinear evolution differential inclusion x ′ ( t ) ∈ A ( t ) x ( t ) + F ( t , x ( t ) ) , where { A ( t ) } t ∈ [ 0 , b ] is a family of linear operators (not necessarily bounded) in a Banach space E generating an evolution operator and F is a Caratheodory type multifunction. First a theorem on the compactness of the set of all mild solutions for the problem is given. Then this result is applied to obtain the existence of mild solutions for the impulsive Cauchy problem defined on non-compact domains.
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TL;DR: In this article, a solution set of systems of equations over a partially commutative group (right-angled Artin group) is given, where hom(H, G) is a finitely generated group.
Abstract: Version 2: Corrected Section 3.3: instead of lexicographical normal forms we now use a normal form due to V. Diekert and A. Muscholl. Consequent changes made and some misprints corrected.
Using an analogue of Makanin-Razborov diagrams, we give an effective description of the solution set of systems of equations over a partially commutative group (right-angled Artin group) $G$. Equivalently, we give a parametrisation of $Hom(H, G)$, where $H$ is a finitely generated group.
TL;DR: In this article, the authors focus on the perturbations of the right and the left side of a linear system and determine the regularity modulus of the solution set mapping, or equivalently on the metric regularity of its inverse mapping.
12 Jul 2008
TL;DR: A method to extract structural information from Pareto-set approximations which offers the possibility to present and visualize the trade-off surface in a compressed form and can be highly useful to reveal hidden structures in compromise solution sets.
Abstract: In a multiobjective setting, evolutionary algorithms can be used to generate a set of compromise solutions. This makes decision making easier for the user as he has alternative solutions at hand which he can directly compare. However, if the number of solutions and the number of decision variables which define the solutions are large, such an analysis may be difficult and corresponding tools are desirable to support a human in separating relevant from irrelevant information.In this paper, we present a method to extract structural information from Pareto-set approximations which offers the possibility to present and visualize the trade-off surface in a compressed form. The main idea is to identify modules of decision variables that are strongly related to each other. Thereby, the set of decision variables can be reduced to a smaller number of significant modules. Furthermore, at the same time the solutions are grouped in a hierarchical manner according to their module similarity. Overall, the output is a dendrogram where the leaves are the solutions and the nodes are annotated with modules. As will be shown on knapsack problem instances and a network processor design application, this method can be highly useful to reveal hidden structures in compromise solution sets.
TL;DR: In this article, a guaranteed interval observer is proposed for nonlinear uncertain continuous-time systems in a bounded-error context, based on Muller's theorem for bracketing the solution of ordinary differential equations in a way which ensures the positivity of the observation error.
TL;DR: In this paper, the weak Pareto optimal solution set of a piecewise linear optimization problem with the ordering cone being convex and having a nonempty interior was shown to be the union of finitely many polyhedra and this set is also arcwise connected under the cone convexity assumption.
Abstract: In general normed spaces, we consider a multiobjective piecewise linear optimization problem with the ordering cone being convex and having a nonempty interior. We establish that the weak Pareto optimal solution set of such a problem is the union of finitely many polyhedra and that this set is also arcwise connected under the cone convexity assumption of the objective function. Moreover, we provide necessary and sufficient conditions about the existence of weak (sharp) Pareto solutions.
TL;DR: In this article, the authors obtained some stability results for the dual problem of a weak vector variational inequality problem and established the upper semicontinuity property of the solution set.
Abstract: In this paper, we obtain some stability results for the dual
problem of a weak vector variational inequality problem. We
establish the upper semicontinuity property of the solution set
for a perturbed dual problem of a weak vector variational
inequality problem. By virtue of a parametric gap function and a
key assumption, we also obtain the lower semicontinuity property
of the solution set for the perturbed dual problem. Some examples
are given for the illustration of the necessity of our research on
duality.
TL;DR: In this article, the boundary of the solution set 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 varying within interval bounds, is characterized by means of pieces of parametric hypersurfaces, the latter represented by their coordinate functions depending on corresponding subsets of n-1 parameters.
Abstract: 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.
TL;DR: This work presents an explicit necessary and sufficient characterization of the symmetric and skew-symmetric solution set by means of nonlinear inequalities.
Abstract: We consider a linear system $Ax=b$, where $A$ is varying inside a given interval matrix $A$, and $b$ is varying inside a given interval vector $b$. The solution set of such a system is described by the well-known Oettli-Prager Theorem. But if we are restricted only on symmetric/skew-symmetric matrices $A\in\mathbf{A}$, the problem is much more complicated. So far, the symmetric/skew-symmetric solution set description could be obtained only by a lengthy Fourier-Motzkin elimination applied on each orthant. We present an explicit necessary and sufficient characterization of the symmetric and skew-symmetric solution set by means of nonlinear inequalities. The number of the inequalities is, however, still exponential w.r.t. the problem dimension.
TL;DR: In this article, the generalized anti-reflexive solution for matrix equations (BX = C, XD = E), which arise in left and right inverse eigenpairs problem, is considered.
Abstract: In this paper, the generalized anti-reflexive solution for matrix equations (BX = C, XD = E), which arise in left and right inverse eigenpairs problem, is considered. With the special properties of generalized anti-reflexive matrices, the necessary and sufficient conditions for the solvability and a general expression of the solution are obtained. Furthermore, the related optimal approximation problem to a given matrix over the solution set is solved. In addition, the algorithm and the example to obtain the unique optimal approximation solution are given.
TL;DR: In this paper, it was shown that for algebraic equations with solutions restricted to large subsets of a finite field, the character sum estimates proved in Part I of this paper are correct.
Abstract: Generalizing earlier results, it is shown that if $$
\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D}
$$
are “large” subsets of a finite field F
q
, then the equations a + b = cd, resp. ab + 1 = cd can be solved with $$
a \in \mathcal{A}, b \in \mathcal{B}, c \in \mathcal{C}, d \in \mathcal{D}
$$
. Other algebraic equations with solutions restricted to “large” subsets of F
q
are also studied. The proofs are based on character sum estimates proved in Part I of the paper.