An important issue in multiobjective optimization is the quantitative comparison of the performance of different algorithms. In the case of multiobjective evolutionary algorithms, the outcome is usually an approximation of the Pareto-optimal set, which is denoted as an approximation set, and therefore the question arises of how to evaluate the quality of approximation sets. Most popular are methods that assign each approximation set a vector of real numbers that reflect different aspects of the quality. Sometimes, pairs of approximation sets are also considered. In this study, we provide a rigorous analysis of the limitations underlying this type of quality assessment. To this end, a mathematical framework is developed which allows one to classify and discuss existing techniques.

/pdf/performance-assessment-of-multiobjective-optimizers-an-wf8l3c8h73.pdf

Performance assessment of multiobjective optimizers: an analysis and review

https://www.mansci.ovgu.de/mansci_media/publikationen/2007/typology-EGOTEC-5t0pvr6fjifln4r4oav60tt612.pdf

An improved typology of cutting and packing problems

The IEEE 802.11 Wireless LAN standards allow multiple non-overlapping frequency channels to be used simultaneously to increase the aggregate bandwidth available to end-users. Such bandwidth aggregation capability is routinely used in infrastructure mode operation, where the traffic to and from wireless nodes is distributed among multiple interfaces of an access point or among multiple access points to balance the traffic load. However, bandwidth aggregation is rarely used in the context of multi-hop 802.11-based LANs that operate in the ad hoc mode. Most past research efforts that attempt to exploit multiple radio channels require modifications to the MAC protocol and therefore do not work with commodity 802.11 interface hardware. In this paper, we propose and evaluate one of the first multi-channel multi-hop wireless ad-hoc network architectures that can be built using standard 802.11 hardware by equipping each node with multiple network interface cards (NICs) operating on different channels. We focus our attention on wireless mesh networks that serve as the backbone for relaying end-user traffic from wireless access points to the wired network. The idea of exploiting multiple channels is particularly appealing in wireless mesh networks because of their high capacity requirements to support backbone traffic. To reap the full performance potential of this architecture, we develop a set of centralized channel assignment, bandwidth allocation, and routing algorithms for multi-channel wireless mesh networks. A detailed performance evaluation shows that with intelligent channel and bandwidth assignment, equipping every wireless mesh network node with just 2 NICs operating on different channels can increase the total network goodput by a factor of up to 8 compared with the conventional single-channel ad hoc network architecture.

Centralized channel assignment and routing algorithms for multi-channel wireless mesh networks

Over the past few years, the research on evolutionary algorithms has demonstrated their niche in solving multiobjective optimization problems, where the goal is to find a number of Pareto-optimal solutions in a single simulation run. Many studies have depicted different ways evolutionary algorithms can progress towards the Pareto-optimal set with a widely spread distribution of solutions. However, none of the multiobjective evolutionary algorithms (MOEAs) has a proof of convergence to the true Pareto-optimal solutions with a wide diversity among the solutions. In this paper, we discuss why a number of earlier MOEAs do not have such properties. Based on the concept of e-dominance, new archiving strategies are proposed that overcome this fundamental problem and provably lead to MOEAs that have both the desired convergence and distribution properties. A number of modifications to the baseline algorithm are also suggested. The concept of e-dominance introduced in this paper is practical and should make the proposed algorithms useful to researchers and practitioners alike.

/pdf/combining-convergence-and-diversity-in-evolutionary-923dko2y76.pdf

Combining convergence and diversity in evolutionary multiobjective optimization

Complexity and Approximation

Approximation algorithms for knapsack problems with cardinality constraints

Semi on-line algorithms for the partition problem

One of the main advantages of portfolios over single assets is that risk can be diversified without necessarily reducing the expected return - provided "proper" assets are selected and they are assigned the "proper" weights. Since in practice investors tend to restrict themselves to a rather small number of different assets, the decision which securities to include is a crucial one that turns out to be NP-hard.

Optimization of cardinality constrained portfolios with a hybrid local search algorithm

The reader may have noticed that for all the considered variants of the knapsack problem, no polynomial time algorithm have been presented which solves the problem to optimality. Indeed all the algorithms described are based on some kind of search and prune methods, which in the worst case may take exponential time. It would be a satisfying result if we somehow could prove it is not possible to find an algorithm which runs in polynomial time, somehow having evidence that the presented methods are “as good as we can do”. However, no proof has been found showing that the considered variants of the knapsack problem cannot be solved to optimality in polynomial time.

Introduction to NP-Completeness of Knapsack Problems

The original Markowitz model of portfolio selection has received a widespread theoretical acceptance and it has been the basis for various portfolio selection techniques. Nevertheless, this normative model has found relatively little application in practice when some additional features, such as fixed costs and minimum transaction lots, are relevant in the portfolio selection problem. In this paper different mixed-integer linear programming models dealing with fixed costs and possibly minimum lots are introduced. Due to the high computational complexity of the models, heuristic procedures, based on the construction and optimal solution of mixed integer subproblems, are proposed. Computational results obtained using data from the Milan Stock Exchange show how the proposed heuristics yield very good solutions in a short computational time and make possible some interesting financial conclusions on the impact of fixed costs and minimum lots on portfolio composition.

Hans Kellerer

Papers

Approximation algorithms for knapsack problems with cardinality constraints

Semi on-line algorithms for the partition problem

Optimization of cardinality constrained portfolios with a hybrid local search algorithm

Introduction to NP-Completeness of Knapsack Problems

Selecting Portfolios with Fixed Costs and Minimum Transaction Lots