Pareto Optimality Notion Research Articles

On cone-based decompositions of proper Pareto-optimality in multi-objective optimization

In recent years, research focus in multi-objective optimization has shifted from approximating the Pareto optimal front in its entirety to identifying solutions that are well-balanced among their objectives. Proper Pareto optimality is an established concept for eliminating Pareto optimal solutions that exhibit unbounded tradeoffs. Imposing a strict tradeoff bound in a classical definition of proper Pareto optimality allows specifying how many units of one objective one is willing to trade in for obtaining one unit of another objective. Recent studies have shown that this notion shows favorable convergence properties. One of the aims of this paper is to translate the proper Pareto optimality notion to an ordering relation, which we denote by M-domination. The mathematical properties of M-domination are thoroughly analyzed in this paper yielding key insights into its applicability as decision making aid and in designing population-based algorithms for solving multi-objective optimization problems. We complement our work by providing four different geometrical descriptions of the M-dominated space given by a union of polyhedral cones. A geometrical description does not only yield a greater understanding of the underlying tradeoff concept, but also allows a quantification of the hypervolume dominated by a particular solution or an entire set of solutions. These descriptions shall enable researchers to formulate hypervolume-based approaches for finding approximations of the Pareto front that emphasize regions that are well-balanced among their tradeoffs in subsequent works.

Identifying and quantifying trade-offs in multi-stakeholder risk evaluation with applications to the data protection impact assessment of the GDPR

Cybersecurity risk management consists of several steps including the selection of appropriate controls to minimize risks. This is a difficult task that requires to search through all possible subsets of a set of available controls and identify those that minimize the risks of all stakeholders. Since stakeholders may have different perceptions of the risks (especially when considering the impact of threats), conflicting goals may arise that require to find the best possible trade-offs among the various needs. In this work, we propose a quantitative and (semi-)automated approach to solve this problem based on the well-known notion of Pareto optimality. For validation, we show how a prototype tool based on our approach can assist in the Data Protection Impact Assessment mandated by the General Data Protection Regulation on a simplified—but realistic—use case scenario. We also evaluate the scalability of the approach by conducting an experimental evaluation with the prototype with encouraging results.

Fair and Efficient Allocations of Chores under Bivalued Preferences

We study the problem of fair and efficient allocation of a set of indivisible chores to agents with additive cost functions. We consider the popular fairness notion of envy-freeness up to one good (EF1) with the efficiency notion of Pareto-optimality (PO). While it is known that EF1+PO allocations exists and can be computed in pseudo-polynomial time in the case of goods, the same problem is open for chores. Our first result is a strongly polynomial-time algorithm for computing an EF1+PO allocation for bivalued instances, where agents have (at most) two disutility values for the chores. To the best of our knowledge, this is the first non-trivial class of chores to admit an EF1+PO allocation and an efficient algorithm for its computation. We also study the problem of computing an envy-free (EF) and PO allocation for the case of divisible chores. While the existence of EF+PO allocation is known via competitive equilibrium with equal incomes, its efficient computation is open. Our second result shows that for bivalued instances, an EF+PO allocation can be computed in strongly polynomial-time.

Collective choice with heterogeneous time preferences

AbstractThis paper reviews recent research on the aggregation of heterogeneous time preferences. Main results are illustrated in a simple Ramsey model with many agents with additively time‐separable utility functions who differ in their discount factors. We employ an intertemporal view on this model and argue that preferences of a decision maker should be represented by a sequence of utility functions. This allows us to clarify the issue of dynamic inconsistency and relate it to simple properties of discounting. We distinguish between private and common consumption cases. In the private consumption case, we discuss the properties of sequences of additively time‐separable social welfare functions (SWFs) and explain why the notion of Pareto optimality under heterogeneous time preferences becomes problematic. In the common consumption case, we focus on the problem of collective choice under heterogeneous time preferences, discuss the difficulties with dynamic voting procedures, and review some ways to overcome them. We conclude by highlighting the implications of our discussion for the problem of choosing an appropriate social discount rate (SDR).

Efficient fair principal component analysis

It has been shown that dimension reduction methods such as Principal Component Analysis (PCA) may be inherently prone to unfairness and treat data from different sensitive groups such as race, color, sex, etc., unfairly. In pursuit of fairness-enhancing dimensionality reduction, using the notion of Pareto optimality, we propose an adaptive first-order algorithm to learn a subspace that preserves fairness, while slightly compromising the reconstruction loss. Theoretically, we provide sufficient conditions that the solution of the proposed algorithm belongs to the Pareto frontier for all sensitive groups; thereby, the optimal trade-off between overall reconstruction loss and fairness constraints is guaranteed. We also provide the convergence analysis of our algorithm and show its efficacy through empirical studies on different datasets, which demonstrates superior performance in comparison with state-of-the-art algorithms. The proposed fairness-aware PCA algorithm can be efficiently generalized to multiple group sensitive features and effectively reduce the unfairness decisions in downstream tasks such as classification.

Collective Choice with Heterogeneous Time Preferences

This paper reviews recent research on the aggregation of heterogeneous time preferences. Main results are illustrated in simple Ramsey models with two or three agents who differ in their discount factors. We employ an intertemporal view on these models and argue that preferences of a decision maker should be represented by a sequence of utility functions. This allows us to clarify the issue of dynamic inconsistency and relate it to simple properties of discounting. We distinguish between private and common consumption cases. In the private consumption case, we discuss the properties of sequences of Paretian social welfare functions and explain why the notion of Pareto optimality under heterogeneous time preferences becomes problematic. In the common consumption case, we focus on the problem of collective choice under heterogeneous time preferences, discuss the difficulties with dynamic voting procedures and review some ways to overcome them. We conclude by highlighting the implications of our discussion for the problem of choosing an appropriate social discount rate.

On Pareto-optimality in the cross-efficiency evaluation

Uniqueness and Pareto-optimality of cross-efficiency scores are two main issues in the cross-efficiency evaluation. We address the Pareto-optimality issue in the cross-efficiency evaluation by presenting a natural notion of Pareto-optimality in the cross-efficiency evaluation which is based on a new self-prioritizing principle and aligned with the concept of dominance. This Pareto-optimality notion is devised using less stringent postulates, compared to those used to devise the existing Pareto-optimality notions in the current literature. We then propose a multi-objective model whose Pareto-optimal solutions generate all the Pareto-optimal cross-efficiency scores sets that fulfill the self-prioritizing principle. It is shown that there is a Pareto-optimal solution to the proposed multi-objective programming model which assigns a common set of weights to all decision making units. We then propose a linear model to obtain such an optimal solution using the weighted sum technique, thereby we can determine an optimal weights profile to generate a set of Pareto-optimal cross-efficiency scores by solving only one linear model. Having started with less stringent postulates, we have more flexibility in enhancing cross-efficiency scores of decision making units and can hence provide a non-dominated set of cross-efficiency scores as illustrated by a real data example. We also demonstrate the possibility of better, compared to other cross-efficiency evaluation methods, resource allocation using another real data example.

Market versus government failures under risk and under uncertainty

Well-established results in the mainstream theory of economic policy show that under assessable risk and asymmetric information, selfish policymakers generate inefficiencies that are generally more harmful than market ones. Market failures are hence an insufficient condition to justify government activism. This conclusion motivates the article’s attempt to understand whether the same conclusion applies under fundamental uncertainty, where selfish policymakers cannot be conceived as maximising inter-temporal expected utility (or any other objective function) calculated by the use of objective probability distributions. After summarising the traditional literature, I stress the difficulty of employing the notion of Pareto optimality in this environment. I then argue that group decision-making and mutual monitoring support <xref ref-type="bibr" rid="CIT0039">Forte’s (1967)</xref> and <xref ref-type="bibr" rid="CIT0029">Demsetz’s (1969)</xref> early claims that no superiority of one institution (market versus government) over the other can a priori be established. The conclusion that the government cannot generally improve on market outcomes does not hence apply.

Computing and testing Pareto optimal committees

Selecting a set of alternatives based on the preferences of agents is an important problem in committee selection and beyond. Among the various criteria put forth for desirability of a committee, Pareto optimality is a minimal and important requirement. As asking agents to specify their preferences over exponentially many subsets of alternatives is practically infeasible, we assume that each agent specifies a weak order on single alternatives, from which a preference relation over subsets is derived using some preference extension. We consider five prominent extensions (responsive, downward lexicographic, upward lexicographic, best, and worst). For each of them, we consider the corresponding Pareto optimality notion, and we study the complexity of computing and verifying Pareto optimal outcomes. For each of the preference extensions, we give a complete characterization of the complexity of testing Pareto optimality when preferences are dichotomous or linear. We also consider strategic issues: for four of the set extensions, we present a linear-time, Pareto optimal and strategyproof algorithm that even works for weak preferences.

The blockwise coordinate descent method for integer programs

Blockwise coordinate descent methods have a long tradition in continuous optimization and are also frequently used in discrete optimization under various names. New interest in blockwise coordinate descent methods arises for improving sequential solutions for problems which consist of several planning stages. In this paper we systematically formulate and analyze the blockwise coordinate descent method for integer programming problems. We discuss convergence of the method and properties of the resulting solutions. We extend the notion of Pareto optimality for blockwise coordinate descent to the case that the blocks do not form a partition and compare Pareto optimal solutions to blockwise optimal and to global optimal solutions. Among others we derive a condition which ensures that the solution obtained by blockwise coordinate descent is weakly Pareto optimal and we confirm convergence of the blockwise coordinate descent to a global optimum in matroid polytopes. The results are interpreted in the context of multi-stage linear integer programming problems and illustrated for integrated planning in public transportation.

On the Pareto Compliance of the Averaged Hausdorff Distance as a Performance Indicator

The averaged Hausdorff distance ∆p is an inframetric, recently introduced in evolutionary multiobjective optimization (EMO) as a tool to measure the optimality of finite size approximations to the Pareto front associated to a multiobjective optimization problem (MOP). Tools of this kind are called performance indicators, and their quality depends on the useful criteria they provide to evaluate the suitability of different candidate solutions to a given MOP. We present here a purely theoretical study of the compliance of the ∆p -indicator to the notion of Pareto optimality. Since ∆p is defined in terms of a modified version of other well- known indicators, namely the generational distance GDp , and the inverted generational distance IGDp , specific criteria for the Pareto compliance of each one of them is discussed in detail. In doing so, we review some previously available knowledge on the behavior of these indicators, correcting inaccuracies found in the literature, and establish new and more general results, including detailed proofs and examples of illustrative situations.

Pareto-optimal reinsurance policies in the presence of individual risk constraints

The notion of Pareto optimality is commonly employed to formulate decisions that reconcile the conflicting interests of multiple agents with possibly different risk preferences. In the context of a one-period reinsurance market comprising an insurer and a reinsurer, both of which perceive risk via distortion risk measures, also known as dual utilities, this article characterizes the set of Pareto-optimal reinsurance policies analytically and visualizes the insurer–reinsurer trade-off structure geometrically. The search of these policies is tackled by translating it mathematically into a functional minimization problem involving a weighted average of the insurer’s risk and the reinsurer’s risk. The resulting solutions not only cast light on the structure of the Pareto-optimal contracts, but also allow us to portray the resulting insurer–reinsurer Pareto frontier graphically. In addition to providing a pictorial manifestation of the compromise reached between the insurer and reinsurer, an enormous merit of developing the Pareto frontier is the considerable ease with which Pareto-optimal reinsurance policies can be constructed even in the presence of the insurer’s and reinsurer’s individual risk constraints. A strikingly simple graphical search of these constrained policies is performed in the special cases of Value-at-Risk and Tail Value-at-Risk.

Pareto-optimal matching allocation mechanisms for boundedly rational agents

Is the Pareto optimality of matching mechanisms robust to the introduction of boundedly rational behavior? To address this question I define a restrictive and a permissive notion of Pareto optimality and consider the large set of hierarchical exchange mechanisms which contains serial dictatorship as well as Gale’s top trading cycles. Fix a housing problem with boundedly rational agents and a hierarchical exchange mechanism. Consider the set of matchings that arise with all possible assignments of agents to initial endowments in the given mechanism. I show that this set is nested between the sets of Pareto optima according to the restrictive and the permissive notion. These containment relations are generally strict, even when deviations from rationality are minimal. In a similar vein, minimal deviations from rationality suffice for the set of outcomes of Gale’s top trading cycles with all possible initial endowments to differ from the set of outcomes of serial dictatorship with all possible orders of agents as dictators.

Application of the Λ-Monotonicity to the Search for Optimal Solutions in Higher-Dimensional Problems

The notion of Pareto optimality is widely used for solving many practical problems. The notion of Λ-optimality is a generalization of the Pareto optimality; the set of Λ-optimal solutions can be either wider or narrower than the set of Pareto-optimal solutions. In this paper, we generalize some results for Λ-optimal target functions obtained earlier, introduce the notion of a critical set of Λ-optimal solutions, and discuss certain approaches to construction of optimal solutions.

Multi‐Objective Cost‐Detectability in Unfalsified Adaptive Control

AbstractVector‐valued controller cost functions that are solely data‐dependent and reflect multiple objectives of a control system are examined within the framework of unfalsified adaptive control. The notion of Pareto optimality of vector‐valued cost functions and the conditions under which they are cost‐detectable are discussed. A sampled data/discrete‐time Level‐Set controller switching algorithm is investigated which allows for the relaxation of the assumption that the controller cost function be monotonically nondecreasing in time. This opens up the possibility of the use of fading memory cost functions which are nonmonotone. When an active controller is falsified at the current threshold cost level, the Level‐Set switching algorithm replaces it by an effectively unique solution of the weighted Tchebycheff method, thus ensuring the selection of an unfalsified Pareto optimal controller. Theoretical results for convergence and stability of the adaptive system are given. Simulation results validate the use of cost‐detectable multi‐objective cost functions. An example of a cost‐detectable cost function which uses fading memory norm of the fictitious tracking error as a performance measure is shown. This allows for computation of performance of nonactive controllers with respect to a reference model.

A unified methodology for single- and multiobjective in-core fuel management optimisation based on augmented Chebyshev scalarisation and a harmony search algorithm

The in-core fuel management optimisation (ICFMO) problem is the problem of finding an optimal fuel reload configuration for a nuclear reactor core. ICFMO may involve the pursuit of a single or multiple objectives, while satisfying several constraints. Very little multiobjective ICFMO research involving the fundamental notion of Pareto optimality has, however, been performed. In this paper, a unified methodology is proposed for the modelling and solution of single- and multiobjective ICFMO problems, be they constrained or unconstrained. With this methodology, ICFMO problems incorporating a variety of objectives and/or constraints may be modelled and solved rapidly, thus providing a cycle-to-cycle optimisation decision support capability for nuclear reactors. An augmented Chebyshev scalarising objective function is incorporated in the methodology for modelling any number of objectives, while an additive penalty function handles potential constraints. Furthermore, an adapted harmony search algorithm is used to solve a given ICFMO problem. The algorithm is able to yield a single solution or a nondominated set of solutions as result (depending on the number of objectives in a problem). The applicability of the methodology is demonstrated by solving (approximately) a variety of ICFMO test problems for the SAFARI-1 nuclear research reactor. The results indicate that the methodology may be used as an effective decision support tool for reactor operators tasked with designing reload configurations from cycle to cycle.

A note on the McKelvey uncovered set and Pareto optimality

We consider the notion of Pareto optimality under the assumption that only the pairwise majority relation is known and show that the set of necessarily Pareto optimal alternatives coincides with the McKelvey uncovered set. As a consequence, the McKelvey uncovered set constitutes the coarsest Pareto optimal majoritarian social choice function. Moreover, every majority relation is induced by a preference profile in which the uncovered alternatives precisely coincide with the Pareto optimal ones. We furthermore discuss the structure of the McKelvey covering relation and the McKelvey uncovered set.

A Data‐Driven Framework for Visual Crowd Analysis

AbstractWe present a novel approach for analyzing the quality of multi‐agent crowd simulation algorithms. Our approach is data‐driven, taking as input a set of user‐defined metrics and reference training data, either synthetic or from video footage of real crowds. Given a simulation, we formulate the crowd analysis problem as an anomaly detection problem and exploit state‐of‐the‐art outlier detection algorithms to address it. To that end, we introduce a new framework for the visual analysis of crowd simulations. Our framework allows us to capture potentially erroneous behaviors on a per‐agent basis either by automatically detecting outliers based on individual evaluation metrics or by accounting for multiple evaluation criteria in a principled fashion using Principle Component Analysis and the notion of Pareto Optimality. We discuss optimizations necessary to allow real‐time performance on large datasets and demonstrate the applicability of our framework through the analysis of simulations created by several widely‐used methods, including a simulation from a commercial game.

Necessary and Sufficient Conditions for Pareto Optimality in Infinite Horizon Cooperative Differential Games

In this technical note, we derive necessary and sufficient conditions for the existence of Pareto solutions in infinite horizon cooperative differential games with open loop information structure. First, we reformulate the notion of Pareto optimality which entails to solving several constrained optimal control problems with a specific constraint structure. Then, we provide weak sufficient conditions under which all Pareto candidates can be obtained by solving a weighted sum optimal control problem, with weightvectors belonging to the unit simplex. We supplement our results with relevant examples.

Unsupervised feature construction for improving data representation and semantics

Feature-based format is the main data representation format used by machine learning algorithms. When the features do not properly describe the initial data, performance starts to degrade. Some algorithms address this problem by internally changing the representation space, but the newly-constructed features are rarely comprehensible. We seek to construct, in an unsupervised way, new features that are more appropriate for describing a given dataset and, at the same time, comprehensible for a human user. We propose two algorithms that construct the new features as conjunctions of the initial primitive features or their negations. The generated feature sets have reduced correlations between features and succeed in catching some of the hidden relations between individuals in a dataset. For example, a feature like $sky \wedge \neg building \wedge panorama$ would be true for non-urban images and is more informative than simple features expressing the presence or the absence of an object. The notion of Pareto optimality is used to evaluate feature sets and to obtain a balance between total correlation and the complexity of the resulted feature set. Statistical hypothesis testing is used in order to automatically determine the values of the parameters used for constructing a data-dependent feature set. We experimentally show that our approaches achieve the construction of informative feature sets for multiple datasets.