308 research outputs found
Computing the hull number in toll convexity
A walk W between vertices u and v of a graph G is called a tolled walk between u and v if u, as well as v, has exactly one neighbour in W. A set S ⊆ V (G) is toll convex if the vertices contained in any tolled walk between two vertices of S are contained in S. The toll convex hull of S is the minimum toll convex set containing S. The toll hull number of G is the minimum cardinality of a set S such that the toll convex hull of S is V (G). The main contribution of this work is an algorithm for computing the toll hull number of a general graph in polynomial time
Fuzzy Bilevel Optimization
In the dissertation the solution approaches for different fuzzy optimization problems are presented. The single-level optimization problem with fuzzy objective is solved by its reformulation into a biobjective optimization problem. A special attention is given to the computation of the membership function of the fuzzy solution of the fuzzy optimization problem in the linear case. Necessary and sufficient optimality conditions of the the convex nonlinear fuzzy optimization problem are derived in differentiable and nondifferentiable cases. A fuzzy optimization problem with both fuzzy objectives and constraints is also investigated in the thesis in the linear case. These solution approaches are applied to fuzzy bilevel optimization problems.
In the case of bilevel optimization problem with fuzzy objective functions, two algorithms are presented and compared using an illustrative example. For the case of fuzzy linear bilevel optimization problem with both fuzzy objectives and constraints k-th best algorithm is adopted.:1 Introduction 1
1.1 Why optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Fuzziness as a concept . . . . . . . . . . . . . . . . . . . . .. . . . . . . 2
1.3 Bilevel problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Preliminaries 11
2.1 Fuzzy sets and fuzzy numbers . . . . . . . . . . . . . . . . . . . . . 11
2.2 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Fuzzy order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Fuzzy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
3 Optimization problem with fuzzy objective 19
3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Solution method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Local optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Existence of an optimal solution . . . . . . . . . . . . . . . . . . . . 25
4 Linear optimization with fuzzy objective 27
4.1 Main approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 Optimality conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.4 Membership function value . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4.1 Special case of triangular fuzzy numbers . . . . . . . . . . . . 36
4.4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39
5 Optimality conditions 47
5.1 Differentiable fuzzy optimization problem . . . . . . . . . . .. . . . 48
5.1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.1.2 Necessary optimality conditions . . . . . . . . . . . . . . . . . . .. 49
5.1.3 Suffcient optimality conditions . . . . . . . . . . . . . . . . . . . . . . 49
5.2 Nondifferentiable fuzzy optimization problem . . . . . . . . . . . . 51
5.2.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2.2 Necessary optimality conditions . . . . . . . . . . . . . . . . . . . 52
5.2.3 Suffcient optimality conditions . . . . . . . . . . . . . . . . . . . . . . 54
5.2.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6 Fuzzy linear optimization problem over fuzzy polytope 59
6.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.2 The fuzzy polytope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63
6.3 Formulation and solution method . . . . . . . . . . . . . . . . . . .. . 65
6.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7 Bilevel optimization with fuzzy objectives 73
7.1 General formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7.2 Solution approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74
7.3 Yager index approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
7.4 Algorithm I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7.5 Membership function approach . . . . . . . . . . . . . . . . . . . . . . .78
7.6 Algorithm II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .80
7.7 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
8 Linear fuzzy bilevel optimization (with fuzzy objectives and constraints) 87
8.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
8.2 Solution approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
8.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
8.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
9 Conclusions 95
Bibliography 9
Exploiting Symmetry in Integer Convex Optimization using Core Points
We consider convex programming problems with integrality constraints that are
invariant under a linear symmetry group. To decompose such problems we
introduce the new concept of core points, i.e., integral points whose orbit
polytopes are lattice-free. For symmetric integer linear programs we describe
two algorithms based on this decomposition. Using a characterization of core
points for direct products of symmetric groups, we show that prototype
implementations can compete with state-of-the-art commercial solvers, and solve
an open MIPLIB problem.Comment: 15 pages; small changes according to suggestions of a referee; to
appear in Operations Research Letter
The Toll Walk Transit Function of a Graph: Axiomatic Characterizations and First-Order Non-definability
A walk , , is called a toll walk if
and and are the only neighbors of and ,
respectively, on in a graph . A toll walk interval , , contains all the vertices that belong to a toll walk between and
. The toll walk intervals yield a toll walk transit function . We represent several axioms that characterize the
toll walk transit function among chordal graphs, trees, asteroidal triple-free
graphs, Ptolemaic graphs, and distance hereditary graphs. We also show that the
toll walk transit function can not be described in the language of first-order
logic for an arbitrary graph.Comment: 31 pages, 4 figures, 25 reference
Minimizing the externalities variance in a LCFS-PR queue under various constraints
Consider a LCFS-PR queue and assume that at time , there are
customers who arrived in that order such that is the arrival time of . Then, the externalities which are generated by
is the total waiting time that would be saved by if
reduced his service requirement to zero. Motivated by some applications,
this work is about the minimization of the externalities variance under various
constraints
Simple and Robust Dynamic Two-Dimensional Convex Hull
The convex hull of a data set is the smallest convex set that contains
.
In this work, we present a new data structure for convex hull, that allows
for efficient dynamic updates. In a dynamic convex hull implementation, the
following traits are desirable: (1) algorithms for efficiently answering
queries as to whether a specified point is inside or outside the hull, (2)
adhering to geometric robustness, and (3) algorithmic simplicity.Furthermore, a
specific but well-motivated type of two-dimensional data is rank-based data.
Here, the input is a set of real-valued numbers where for any number its rank is its index in 's sorted order. Each value in can be mapped
to a point to obtain a two-dimensional point set. In this work,
we give an efficient, geometrically robust, dynamic convex hull algorithm, that
facilitates queries to whether a point is internal. Furthermore, our
construction can be used to efficiently update the convex hull of rank-ordered
data, when the real-valued point set is subject to insertions and deletions.
Our improved solution is based on an algorithmic simplification of the
classical convex hull data structure by Overmars and van Leeuwen~[STOC'80],
combined with new algorithmic insights. Our theoretical guarantees on the
update time match those of Overmars and van Leeuwen, namely ,
while we allow a wider range of functionalities (including rank-based data).
Our algorithmic simplification includes simplifying an 11-case check down to a
3-case check that can be written in 20 lines of easily readable C-code. We
extend our solution to provide a trade-off between theoretical guarantees and
the practical performance of our algorithm. We test and compare our solutions
extensively on inputs that were generated randomly or adversarially, including
benchmarking datasets from the literature.Comment: Accepted for ALENEX2
Low rank representations of matrices using nuclear norm heuristics
2014 Summer.The pursuit of low dimensional structure from high dimensional data leads in many instances to the finding the lowest rank matrix among a parameterized family of matrices. In its most general setting, this problem is NP-hard. Different heuristics have been introduced for approaching the problem. Among them is the nuclear norm heuristic for rank minimization. One aspect of this thesis is the application of the nuclear norm heuristic to the Euclidean distance matrix completion problem. As a special case, the approach is applied to the graph embedding problem. More generally, semi-definite programming, convex optimization, and the nuclear norm heuristic are applied to the graph embedding problem in order to extract invariants such as the chromatic number, Rn embeddability, and Borsuk-embeddability. In addition, we apply related techniques to decompose a matrix into components which simultaneously minimize a linear combination of the nuclear norm and the spectral norm. In the case when the Euclidean distance matrix is the distance matrix for a complete k-partite graph it is shown that the nuclear norm of the associated positive semidefinite matrix can be evaluated in terms of the second elementary symmetric polynomial evaluated at the partition. We prove that for k-partite graphs the maximum value of the nuclear norm of the associated positive semidefinite matrix is attained in the situation when we have equal number of vertices in each set of the partition. We use this result to determine a lower bound on the chromatic number of the graph. Finally, we describe a convex optimization approach to decomposition of a matrix into two components using the nuclear norm and spectral norm
Development and implementation of a B-Spline motion planning framework for autonomous mobile robots
O projeto enquadra-se na área da robótica. A ideia deste projeto é utilizar as propriedades das curvas b-spline para resolver problemas de otimização de motion planning. Esta abordagem permite desviar dos tradicionais motion planning algorithms que são normalmente utilizados. Devido á sua natureza matemática, esta abordagem permite a utilização de teoremas como o Separating Hyperplane Thereoem para realizar o desvio de obstáculos. Um aspecto importante a ter em conta é que este projeto irá ser integrado com os projetos desenvolvidos por outros alunos de modo a participar na competição The Autonomous Ship Challenge, a ser realizada na Noruega.This project fits within the area of robotics. The main idea is to utilize the properties of b-splines curves in order to solve motion planning optimization problems. This approach allows to deviate from the traditional motion planning algorithms, that are usually used. Due to its mathematical nature, this approach allows the use of theorems like the Separating Hyperplane Theorem for the obstacle avoidance problem. An important aspect to notice is that this project will be integrated with the other projects developed by other students in order to participate in "The Autonomous Ship Challenge" competition to be held in Norway
Capacity and Price Competition in Markets with Congestion Effects
We study oligopolistic competition in service markets where firms offer a
service to customers. The service quality of a firm - from the perspective of a
customer - depends on the congestion and the charged price. A firm can set a
price for the service offered and additionally decides on the service capacity
in order to mitigate congestion. The total profit of a firm is derived from the
gained revenue minus the capacity investment cost. Firms simultaneously set
capacities and prices in order to maximize their profit and customers
subsequently choose the services with lowest combined cost (congestion and
price). For this basic model, Johari et al. (2010) derived the first existence
and uniqueness results of pure Nash equilibria (PNE) assuming mild conditions
on congestion functions. Their existence proof relies on Kakutani's fixed-point
theorem and a key assumption for the theorem to work is that demand for service
is elastic (modeled by a smooth and strictly decreasing inverse demand
function).
In this paper, we consider the case of perfectly inelastic demand, i.e. there
is a fixed volume of customers requesting service. This scenario applies to
realistic cases where customers are not willing to drop out of the market, e.g.
if prices are regulated by reasonable price caps. We investigate existence,
uniqueness and quality of PNE for models with inelastic demand and price caps.
We show that for linear congestion cost functions, there exists a PNE. This
result requires a completely new proof approach compared to previous
approaches, since the best response correspondences of firms may be empty, thus
standard fixed-point arguments are not directly applicable. We show that the
game is C-secure (see McLennan et al. (2011)), which leads to the existence of
PNE. We furthermore show that the PNE is unique, and that the efficiency
compared to a social optimum is unbounded in general.Comment: A one-page abstract of this paper appeared in the proceedings of the
15th International Conference on Web and Internet Economics (WINE 2019
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