92 research outputs found
Twelve monotonicity conditions arising from algorithms for equilibrium problems
In the last years many solution methods for equilibrium problems (EPs) have been developed. Several different monotonicity conditions have been exploited to prove convergence. The paper investigates all the relationships between them in the framework of the so-called abstract EP. The analysis is further detailed for variational inequalities and linear EPs, which include also Nash EPs with quadratic payoffs
Descent and penalization techniques for equilibrium problems with nonlinear constraints
This paper deals with equilibrium problems with nonlinear constraints. Exploiting a gap function recently introduced, which rely on a polyhedral approximation of the feasible region, we propose two descent methods. They are both based on the minimization of a suitable exact penalty function, but they use different rules for updating the penalization parameter and they rely on different types of line search. The convergence of both algorithms is proved under standard assumptions
A Game-Theoretic Approach for Runtime Capacity Allocation in MapReduce
Nowadays many companies have available large amounts of raw, unstructured
data. Among Big Data enabling technologies, a central place is held by the
MapReduce framework and, in particular, by its open source implementation,
Apache Hadoop. For cost effectiveness considerations, a common approach entails
sharing server clusters among multiple users. The underlying infrastructure
should provide every user with a fair share of computational resources,
ensuring that Service Level Agreements (SLAs) are met and avoiding wastes. In
this paper we consider two mathematical programming problems that model the
optimal allocation of computational resources in a Hadoop 2.x cluster with the
aim to develop new capacity allocation techniques that guarantee better
performance in shared data centers. Our goal is to get a substantial reduction
of power consumption while respecting the deadlines stated in the SLAs and
avoiding penalties associated with job rejections. The core of this approach is
a distributed algorithm for runtime capacity allocation, based on Game Theory
models and techniques, that mimics the MapReduce dynamics by means of
interacting players, namely the central Resource Manager and Class Managers
Gap functions for quasi-equilibria
An approach for solving quasi-equilibrium problems (QEPs) is proposed relying on gap functions, which allow reformulating QEPs as global optimization problems. The (generalized) smoothness properties of a gap function are analysed and an upper estimate of its Clarke directional derivative is given. Monotonicity assumptions on both the equilibrium and constraining bifunctions are a key tool to guarantee that all the stationary points of a gap function actually solve QEP. A few classes of constraints satisfying such assumptions are identified covering a wide range of situations. Relying on these results, a descent method for solving QEP is devised and its convergence proved. Finally, error bounds are given in order to guarantee the boundedness of the sequence generated by the algorithm
Auxiliary problem principles for equilibria
The auxiliary problem principle allows solving a given equilibrium problem (EP) through an equivalent auxiliary problem with better properties. The paper investigates two families of auxiliary EPs: the classical auxiliary problems, in which a regularizing term is added to the equilibrium bifunction, and the regularized Minty EPs. The conditions that ensure the equivalence of a given EP with each of these auxiliary problems are investigated exploiting parametric definitions of different kinds of convexity and monotonicity. This analysis leads to extending some known results for variational inequalities and linear EPs to the general case together with new equivalences. Stationarity and convexity properties of gap functions are investigated as well in this framework. Moreover, both new results on the existence of a unique solution and new error bounds based on gap functions with good convexity properties are obtained under weak quasimonotonicity or weak concavity assumptions
D-gap functions and descent techniques for solving equilibrium problems
A new algorithm for solving equilibrium problems with differentiable bifunctions is provided. The algorithm is based on descent directions of a suitable family of D-gap functions. Its convergence is proved under assumptions which do not guarantee the equivalence between the stationary points of the D-gap functions and the solutions of the equilibrium problem. Moreover, the algorithm does not require to set parameters according to thresholds which depend on regularity properties of the equilibrium bifunction. The results of preliminary numerical tests on Nash equilibrium problems with quadratic payoffs are reported. Finally, some numerical comparisons with other D-gap algorithms are drawn relying on some further tests on linear equilibrium problems
Gap functions for quasi-equilibria
An approach for solving quasi-equilibrium problems (QEPs) is proposed relying on gap functions, which allow reformulating QEPs as global optimization problems. The (generalized) smoothness properties of a gap function are analysed and an upper estimates of its Clarke directional derivative is given.
Monotonicity assumptions on both the equilibrium and constraining bifunctions are a key tool to guarantee that all the stationary points of a gap function actually solve QEP. A few classes of constraints satisfying such assumptions are identified covering a wide range of situations. Relying on these results, a descent method for solving QEP is devised and its convergence proved. Finally, error bounds are given in order to guarantee the boundedness of the sequence generated by the algorithm
Solving non-monotone equilibrium problems via a DIRECT-type approach
A global optimization approach for solving non-monotone equilibrium problems
(EPs) is proposed. The class of (regularized) gap functions is used to
reformulate any EP as a constrained global optimization program and some bounds
on the Lipschitz constant of such functions are provided. The proposed global
optimization approach is a combination of an improved version of the
\texttt{DIRECT} algorithm, which exploits local bounds of the Lipschitz
constant of the objective function, with local minimizations. Unlike most
existing solution methods for EPs, no monotonicity-type condition is assumed in
this paper. Preliminary numerical results on several classes of EPs show the
effectiveness of the approach.Comment: Technical Report of Department of Computer Science, University of
Pisa, Ital
Service provisioning problem in cloud and multi-cloud systems
Cloud computing is a new emerging paradigm that aims to streamline the on-demand provisioning of resources as services, providing end users with flexible and scalable services accessible through the Internet on a pay-per-use basis. Because modern cloud systems operate in an open and dynamic world characterized by continuous changes, the development of efficient resource provisioning policies for cloud-based services becomes increasingly challenging. This paper aims to study the hourly basis service provisioning problem through a generalized Nash game model. We take the perspective of Software as a Service (SaaS) providers that want to minimize the costs associated with the virtual machine instances allocated in a multiple Infrastructures as a Service (IaaS) scenario while avoiding incurring penalties for execution failures and providing quality of service guarantees. SaaS providers compete and bid for the use of infrastructural resources, whereas the IaaSs want to maximize their revenues obtained providing virtualized resources. We propose a solution algorithm based on the best-reply dynamics, which is suitable for a distributed implementation. We demonstrate the effectiveness of our approach by performing numerical tests, considering multiple workloads and system configurations. Results show that our algorithm is scalable and provides significant cost savings with respect to alternative methods (5% on average but up to 260% for individual SaaS providers). Furthermore, varying the number of IaaS providers means an 8%-15% cost savings can be achieved from the workload distribution on multiple IaaSs
Chemotherapy planning and multi-appointment scheduling: formulations, heuristics and bounds
The number of new cancer cases is expected to increase by about 50% in the
next 20 years, and the need for chemotherapy treatments will increase
accordingly. Chemotherapy treatments are usually performed in outpatient cancer
centers where patients affected by different types of tumors are treated. The
treatment delivery must be carefully planned to optimize the use of limited
resources, such as drugs, medical and nursing staff, consultation and exam
rooms, and chairs and beds for the drug infusion. Planning and scheduling
chemotherapy treatments involve different problems at different decision
levels. In this work, we focus on the patient chemotherapy multi-appointment
planning and scheduling problem at an operational level, namely the problem of
determining the day and starting time of the oncologist visit and drug infusion
for a set of patients to be scheduled along a short-term planning horizon. We
use a per-pathology paradigm, where the days of the week in which patients can
be treated, depending on their pathology, are known. We consider different
metrics and formulate the problem as a multi-objective optimization problem
tackled by sequentially solving three problems in a lexicographic
multi-objective fashion. The ultimate aim is to minimize the patient's
discomfort. The problems turn out to be computationally challenging, thus we
propose bounds and ad-hoc approaches, exploiting alternative problem
formulations, decomposition, and -opt search. The approaches are tested on
real data from an Italian outpatient cancer center and outperform
state-of-the-art solvers.Comment: 28 pages, 3 figure
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