Mathematical programs with vanishing constraints: Optimality conditions, sensitivity, and a relaxation method.

*(English)*Zbl 1180.90312The paper of A. F. Izmailov and M. V. Solodov is located in a relatively new area of nonlinear continuous optimization called Mathematical Programs with Vanishing Constraints (MPVQ), certainly influenced by mathematical programs with equilibrium constraints or with complementarity conditions, but inspired also by practical applications. Herewith, and since MPVQs in a natural way generalize mathematical programs with equality constraints - via the use of inequality constraints and a multiplication of equality with inequality kind of functions, this paper extends the more classical theory of mathematical programming very widely. What is more, since the article pays a special attention to a sensitivity and perturbational theory and, already, to a relaxation method for MPVQs, it prepares the way for a deeper analytical and, hence, topological understanding of MPVQs and, by this, for the development of future numerical solution methods (e.g., pathfollowing, interior point or reduction procedures).

In fact, the authors consider MPVQs as a class of programs with switch-off/switch-on constraints (in inequality form). These problems are specific in the sense that constraints can be “switched on” at some feasible points, but “switched off” at others (explaining also the “vanishing contraints”). Practical motivations come from the field of topology design. MPVQs are usually degenerate at solutions, but structurally different from programs with complementarity constraints. The authors discuss well-known first- and second-order necessary optimality conditions for MPVQs, and they arrive at new second-order sufficient optimality conditions. They are equivalent to classical ones from optimization. By these preparations, they put the foundations for sensitivity, perturbation and relaxation studies, which may surely be expected to give and important incentive and support for future developments and applications in engineering, OR, game theory and economics.

In fact, the authors consider MPVQs as a class of programs with switch-off/switch-on constraints (in inequality form). These problems are specific in the sense that constraints can be “switched on” at some feasible points, but “switched off” at others (explaining also the “vanishing contraints”). Practical motivations come from the field of topology design. MPVQs are usually degenerate at solutions, but structurally different from programs with complementarity constraints. The authors discuss well-known first- and second-order necessary optimality conditions for MPVQs, and they arrive at new second-order sufficient optimality conditions. They are equivalent to classical ones from optimization. By these preparations, they put the foundations for sensitivity, perturbation and relaxation studies, which may surely be expected to give and important incentive and support for future developments and applications in engineering, OR, game theory and economics.

Reviewer: Gerhard-Wilhelm Weber (Ankara)

##### MSC:

90C30 | Nonlinear programming |

90C46 | Optimality conditions and duality in mathematical programming |

90C31 | Sensitivity, stability, parametric optimization |

##### Keywords:

Mathematical program with vanishing constraints; Stationarity; Second-order conditions; Sensitivity; Perturbations; Relaxation
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\textit{A. F. Izmailov} and \textit{M. V. Solodov}, J. Optim. Theory Appl. 142, No. 3, 501--532 (2009; Zbl 1180.90312)

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