Multiobjective and stochastic optimization based on parametric optimization.

*(English)*Zbl 0583.90055
Mathematical Research, 26. Berlin: Akademie-Verlag. 175 p. M. 22.00 (1985).

Under quite general assumptions, admissible (efficient) points of multiobjective optimization problems and admissible (efficient with probability 1) points for stochastic programs in the expectation form are closely related to optimal solutions for parametric programs with parameters in the objective function and/or on the right-hand sides (for a survey see chapter 2). These results are used to get algorithmic procedures which provide a possibility to get a sufficiently large set of admissible solutions in a numerically effective way:

As a starting point, algorithms for linear parametric programs involving one parameter in the objective function or/and on the right-hand sides are explained in detail in chapter 3. For differentiable nonlinear parametric programs depending on a scalar parameter, solution methods based on continuation techniques are presented in chapter 4. Dialogue algorithms resulting from the explained parametric optimization techniques are developed in chapter 5; their specification to solving multiobjective optimization problems is given and their applicability for solving special classes of stochastic programs is discussed. In the last chapter, selected examples (optimization of production plans, sample surveys, dynamic transportation problem, system optimizing under stochastic disturbancies) illustrate the practical value of the presented results.

Theoretical results are given partly without proofs, the algorithms are completed by flow charts and numerical examples, and numerous references are attached.

As a starting point, algorithms for linear parametric programs involving one parameter in the objective function or/and on the right-hand sides are explained in detail in chapter 3. For differentiable nonlinear parametric programs depending on a scalar parameter, solution methods based on continuation techniques are presented in chapter 4. Dialogue algorithms resulting from the explained parametric optimization techniques are developed in chapter 5; their specification to solving multiobjective optimization problems is given and their applicability for solving special classes of stochastic programs is discussed. In the last chapter, selected examples (optimization of production plans, sample surveys, dynamic transportation problem, system optimizing under stochastic disturbancies) illustrate the practical value of the presented results.

Theoretical results are given partly without proofs, the algorithms are completed by flow charts and numerical examples, and numerous references are attached.

Reviewer: J.Dupacova

##### MSC:

90C31 | Sensitivity, stability, parametric optimization |

90C15 | Stochastic programming |

90-02 | Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming |

65K05 | Numerical mathematical programming methods |

90C90 | Applications of mathematical programming |