We present our view of the state of the art in continuous multiobjective programming. After an introduction we formulate the multiobjective program (MOP) and define the most important solution concepts in Sect. 18.2. In Sect. 18.3 we summarize properties of efficient and nondominated sets. Optimality conditions are reviewed in Sect. 18.4. The main part of the chapter consists of Sects. 18.5 and 18.6 that deal with solution techniques for MOPs and approximation of efficient and nondominated sets. In Sect. 18.7 we discuss specially-structured problems including linear, nonlinear, parametric, and bilevel MOPs. In Sect. 18.8 we present our perspective on future research directions.

Continuous Multiobjective Programming

We consider a multiobjective linear program and the coefficients of the multiobjective function are supposed to be uncertain Let x* be an efficient point We propose a procedure computing a tolerance for each objective function coefficient, such that all these coefficients may simultaneously and independently vary within their tolerances while preserving the efficiency of x* If x* is a non-degenerate basic solution, then the procedure runs in a polynomial time Our method is also applicable for the intervals of multiobjective linear programming for checking the necessary efficiency of x*, ie whether x* is efficient for all the realizations of interval values

Computing the tolerances in multiobjective linear programming

In general normed spaces, we consider a multiobjective piecewise linear optimization problem with the ordering cone being convex and having a nonempty interior. We establish that the weak Pareto optimal solution set of such a problem is the union of finitely many polyhedra and that this set is also arcwise connected under the cone convexity assumption of the objective function. Moreover, we provide necessary and sufficient conditions about the existence of weak (sharp) Pareto solutions.

The structure of weak Pareto solution sets in piecewise linear multiobjective optimization in normed spaces

In general Banach spaces, we consider a vector optimization problem (SVOP) in which the objective is a set-valued mapping whose graph is the union of finitely many polyhedra. We establish some results on structure and connectedness of the weak Pareto solution set, Pareto solution set, weak Pareto optimal value set and Pareto optimal value set of (SVOP). In particular, we improve and generalize Arrow, Barankin and Blackwell’s classical results on linear vector optimization problems in Euclidean spaces.

Pareto Solutions of Polyhedral-valued Vector Optimization Problems in Banach Spaces

This introductory treatment of the theory and methodology of nonlinear multiobjective programming offers an annotated bibliographic overview of various concepts and approaches to solve continuous deterministic multicriteria optimization problems. First, the notions of Pareto optimality, efficiency, and nondominance are discussed and then related to approximate, epsilon, and proper efficiency. Second, after highlighting similarities and pointing out differences to first- and second-order optimality conditions in classical nonlinear programming, several other topics including duality and sensitivity are also mentioned. Third, a number of solution approaches and generating methods are described with a particular focus on scalarizations using objective aggregation, prioritization, and norm or distance minimization to some utopian reference point.


Keywords:

annotated survey;
nonlinear multiobjective programming;
multicriteria optimization;
vector optimization;
partial orders;
ordering cones;
Pareto optimality;
efficiency;
approximate efficiency;
epsilon efficiency;
proper efficiency;
trade-offs;
efficient solutions;
nondominance;
nondominated points;
optimality conditions;
duality;
sensitivity;
scalarization;
generating methods;
weighted sum;
epsilon-constraint;
Chebyshev norm

Nonlinear Multiobjective Programming

In this paper, we prove that, if the data of a linear multiobjectiveprogramming problem are smooth functions of a parameter, then in theparameter space there is an open dense subset where the efficient solutionset of the problem can be locally represented as a union of some faces whosevertices and directions are smooth functions of the parameter.

On sensitivity in linear multiobjective programming

First-principles study of electronic and optical properties of defective sawtooth penta-graphene nanoribbons

Monolayer germanium sulfide is gaining significant attention for its exceptional anisotropic electronic conductance, notable excitonic effects, and wide range of potential applications. In our study, we used density functional theory, many-body perturbation theory, and non-equilibrium Green function to investigate electronic transport properties and exciton radiative lifetime of single-layer germanium sulfide. Our theoretical findings showed that applying up to 8 percent compressive strain increased carrier mobility by nearly threefold, and thus, dramatically enhance the device's current intensity. Moreover, we observed that strain engineering allowed fine-tuning of the electron-hole recombination time. At 6 percent tensile strain, the effective radiative lifetime was as short as 19 picoseconds, which is 4.5 times faster than the intrinsic state and 80 times faster than at 8 percent compressive strain. These results highlight the potential of strain engineering to customize the electronic and optical properties of GeS monolayer for specific electronic, optoelectronic, and photovoltaic device requirements

L. V. Thuan

Papers

On sensitivity in linear multiobjective programming

First-principles study of electronic and optical properties of defective sawtooth penta-graphene nanoribbons

Unlocking the Potential of GeS Monolayer: Strain-Enabled Control of Electronic Transports and Exciton Radiative Lifetimes