On local Pareto optima

doi:10.1016/0304-4068(75)90012-9

Journal of Mathematical Economics

Volume 2, Issue 1, March 1975, Pages 35-42

https://doi.org/10.1016/0304-4068(75)90012-9 Get rights and content

Abstract

This paper is concerned with the local problem of optimizing several functions at once. The first-order (necessary) conditions for a local Pareto Optimum are well-known. In this paper we obtain the best second-order criteria for local Pareto Optima in an invariant form.

References (7)

H.I. Levine
Singularities of differentiable mappings
Proceedings of the Liverpool Singularities Symposium I
(1971)
V. Pareto
Manual of political economy
(1927)
B.N. Pshenichuyi
Necessary conditions for an extremum
(1971)

There are more references available in the full text version of this article.

Cited by (32)

Peeking beyond peaks: Challenges and research potentials of continuous multimodal multi-objective optimization
2021, Computers and Operations Research
Citation Excerpt :
We will address this important area in Section 5.3 in more detail and from an algorithmic perspective. Note further that we do not strive for reinventing notions of localness — those are already discussed in early theoretical works that deal with local efficiency in MOO and aim for first and second order optimality criteria (e.g., refer to Wan, 1975; Van Geldrop, 1980; Jiménez, 2002). Locally Efficient Point
Multi-objective (MO) optimization, i.e., the simultaneous optimization of multiple conflicting objectives, is gaining more and more attention in various research areas, such as evolutionary computation, machine learning (e.g., (hyper-)parameter optimization), or logistics (e.g., vehicle routing). Many works in this domain mention the structural problem property of multimodality as a challenge from two classical perspectives: (1) finding all globally optimal solution sets, and (2) avoiding to get trapped in local optima. Interestingly, these streams seem to transfer many traditional concepts of single-objective (SO) optimization into claims, assumptions, or even terminology regarding the MO domain, but mostly neglect the understanding of the structural properties as well as the algorithmic search behavior on a problem’s landscape. However, some recent works counteract this trend, by investigating the fundamentals and characteristics of MO problems using new visualization techniques and gaining surprising insights.
Using these visual insights, this work proposes a step towards a unified terminology to capture multimodality and locality in a broader way than it is usually done. This enables us to investigate current research activities in multimodal continuous MO optimization and to highlight new implications and promising research directions for the design of benchmark suites, the discovery of MO landscape features, the development of new MO (or even SO) optimization algorithms, and performance indicators. For all these topics, we provide a review of ideas and methods but also an outlook on future challenges, research potential and perspectives that result from recent developments.
Impact of dynamic wheel load on roadway infrastructure sustainability
2021, Transportation Research Part D: Transport and Environment
Dynamic wheel load (DWL) may result in additional pavement damage. This study examined the impact of DWL on pavement performance using mechanistic tractor-trailer and three-dimensional (3-D) viscoelastic finite element (FE) pavement models. Based on the result of FE simulations, a correlation between the pavement responses and DWL was established. Using the mechanistic-empirical pavement design guide approach, the pavement service life was predicted. A case study based on life cycle assessment (LCA) and life cycle costing (LCC) methodologies was performed to study the DWL impact on greenhouse gas (GHG) emissions, energy consumption, and cost. The study identified the probabilistic distribution of pavement service life due to DWL. The effect of pavement service life on the environmental impact of each LCA stage was investigated. Finally, optimal sustainable solutions considering both environmental and economic impacts were introduced using a Pareto frontier analysis.
Strict efficiency in vector optimization
2002, Journal of Mathematical Analysis and Applications
The notion of strict minimum of order m for real optimization problems is extended to vector optimization. Its properties and characterization are studied in the case of finite-dimensional spaces (multiobjective problems). Also the notion of super-strict efficiency is introduced for multiobjective problems, and it is proved that, in the scalar case, all of them coincide. Necessary conditions for strict minimality and for super-strict minimality of order m are provided for multiobjective problems with an arbitrary feasible set. When the objective function is Fréchet differentiable, necessary and sufficient conditions are established for the case m = 1, resulting in the situation that the strict efficiency and super-strict efficiency notions coincide.
Multiple-objective programming with polynomial objectives and constraints
1992, European Journal of Operational Research
Stability and instability in oligopoly
1986, Journal of Economic Theory
For regular oligopolies (both homogeneous and heterogeneous) the (local) Cournot-Nash equilibria are the (non-degenerate) critical points of a Morse-Smale vector field, defined on the feasible region of non-negative prices and outputs. When at the boundary of this feasible region this vector field points inwards, it follows from the Morse inequalities that there is at least one stable equilibrium. When there is a unique, non-stable, interior equilibrium, necessarily the vector field points outwards somewhere along the boundary of the feasible region. This raises to a stable boundary equilibrium.
On efficient sets in vector maximum problems - A brief survey
1986, European Journal of Operational Research
The notion of an efficient (nondominated, noninferior; Pareto-optimal, functional efficient) set to a Vector Maximum Problem (VMP) has been analyzed and developed in several directions during the last 30 years. Starting with the basic notion of efficiency given by Pareto (1896), formal descriptions of the efficient, properly efficient, locally (proper-) efficient and weak or strong efficient set have been developed. Based on these notions, various characteristics and properties of the efficient set have been studied. The structure of the efficient set and the existence of efficient solutions have also been analyzed based on the various properties of the feasible set X and the functions z_k(x), k = 1,…, K, which constitute the vector-valued criterion z(x) of VMP (e.g., convexity of X, concavity of z_k(x) for all k, differentiability). A portion of the literature is of a rather pure theoretical nature. Other portions try to develop theories to serve as the basis for developing methods for the determination of the efficient set or for developing interactive methods for determination of compromise solutions. Duality theories for more or less general cases have been developed and various aspects of stability of VMP have been investigated. A brief survey is given.

View all citing articles on Scopus

^∗: The author wishes to thank Professor S. Smale for suggesting this problem, and his many valuable comments.

View full text

On local Pareto optima

Abstract

Singularities of differentiable mappings

Proceedings of the Liverpool Singularities Symposium I

Manual of political economy

Necessary conditions for an extremum