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LP-based approximation algorithms for capacitated facility location
In the capacitated facility location problem with hard capacities, we are given a set of facilities,
, and a set of clients
in a common metric space. Each facility
i
has a
facility opening cost
f
i
and
capacity
u
i
that specifies the maximum number of clients that may be assigned to this facility. We want to
open
some facilities from the set
and assign each client to an open facility so that at most
u
i
clients are assigned to any open facility
i
. The cost of assigning client
j
to facility
i
is given by the distance
c
ij
, and our goal is to minimize the sum of the facility opening costs and the client assignment costs. The only known approximation algorithms that deliver solutions within a constant factor of optimal for this NP-hard problem are based on local search techniques. It is an open problem to devise an approximation algorithm for this problem based on a linear programming lower bound (or indeed, to prove a constant integrality gap for any LP relaxation). We make progress on this question by giving a 5-approximation algorithm for the special case in which all of the facility costs are equal, by rounding the optimal solution to the standard LP relaxation. One notable aspect of our algorithm is that it relies on partitioning the input into a collection of single-demand capacitated facility location problems, approximately solving them, and then combining these solutions in a natural way.
Journal Article
Fuzzy Multi-Criteria Decision Making
In summarizing the concepts and results of the most popular fuzzy multicriteria methods, using numerical examples, this work examines all the most recently developed methods. Each one of the 22 chapters include practical applications along with new results.
eBook
Robust Optimization
Robust optimization is still a relatively new approach to optimization problems affected by uncertainty, but it has already proved so useful in real applications that it is difficult to tackle such problems today without considering this powerful methodology. Written by the principal developers of robust optimization, and describing the main achievements of a decade of research, this is the first book to provide a comprehensive and up-to-date account of the subject. Robust optimization is designed to meet some major challenges associated with uncertainty-affected optimization problems: to operate under lack of full information on the nature of uncertainty; to model the problem in a form that can be solved efficiently; and to provide guarantees about the performance of the solution. The book starts with a relatively simple treatment of uncertain linear programming, proceeding with a deep analysis of the interconnections between the construction of appropriate uncertainty sets and the classical chance constraints (probabilistic) approach. It then develops the robust optimization theory for uncertain conic quadratic and semidefinite optimization problems and dynamic (multistage) problems. The theory is supported by numerous examples and computational illustrations. An essential book for anyone working on optimization and decision making under uncertainty, Robust Optimization also makes an ideal graduate textbook on the subject.
eBook
Optimization Algorithms on Matrix Manifolds
Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra. Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists.
eBook
Linear convergence of first order methods for non-strongly convex optimization
The standard assumption for proving linear convergence of first order methods for smooth convex optimization is the strong convexity of the objective function, an assumption which does not hold for many practical applications. In this paper, we derive linear convergence rates of several first order methods for solving smooth non-strongly convex constrained optimization problems, i.e. involving an objective function with a Lipschitz continuous gradient that satisfies some relaxed strong convexity condition. In particular, in the case of smooth constrained convex optimization, we provide several relaxations of the strong convexity conditions and prove that they are sufficient for getting linear convergence for several first order methods such as projected gradient, fast gradient and feasible descent methods. We also provide examples of functional classes that satisfy our proposed relaxations of strong convexity conditions. Finally, we show that the proposed relaxed strong convexity conditions cover important applications ranging from solving linear systems, Linear Programming, and dual formulations of linearly constrained convex problems.
Journal Article
Golden ratio algorithms for variational inequalities
The paper presents a fully adaptive algorithm for monotone variational inequalities. In each iteration the method uses two previous iterates for an approximation of the local Lipschitz constant without running a linesearch. Thus, every iteration of the method requires only one evaluation of a monotone operator F and a proximal mapping g. The operator F need not be Lipschitz continuous, which also makes the algorithm interesting in the area of composite minimization. The method exhibits an ergodic O(1 / k) convergence rate and R-linear rate under an error bound condition. We discuss possible applications of the method to fixed point problems as well as its different generalizations.
Journal Article
Worst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models
The worst-case evaluation complexity for smooth (possibly nonconvex) unconstrained optimization is considered. It is shown that, if one is willing to use derivatives of the objective function up to order
p
(for
p
≥
1
) and to assume Lipschitz continuity of the
p
-th derivative, then an
ϵ
-approximate first-order critical point can be computed in at most
O
(
ϵ
-
(
p
+
1
)
/
p
)
evaluations of the problem’s objective function and its derivatives. This generalizes and subsumes results known for
p
=
1
and
p
=
2
.
Journal Article