Second order optimality conditions
for differentiable multiobjective problems

Giancarlo Bigi and Marco Castellani


The study of optimality conditions is one of the main topics of Optimization Theory. For multiobjective programming, some of the first interesting results have been developed in the middle seventies; since then, many papers appeared, dealing with first order necessary optimality conditions both for differentiable and nondifferentiable problems When the problem satisfies suitable convexity assumptions, these conditions turn out to be also sufficient. However, in the general case there may be feasible points, which satisfy the first order conditions but are not optimal solutions. In order to drop them, additional optimality conditions, involving second order derivatives of the given functions, can be developed. A few results in this direction have been presented in some recent papers. This paper aims to deepen this type of analysis, providing more general results. First, we investigate differentiable multiobjective problems, where the constraint is given in set form. By linearizing tecniques, we obtain necessary conditions in terms of the impossibility of nonhomogeneous linear systems, involving the Jacobians and the Hessians of the objective functions and the second order contingent set of the feasible region. We stress that these systems depend upon the choice of a common descent direction for the objective functions. Moreover, we show that the gap between first order conditions for single and multi objective problems holds also for second order conditions. Then, we apply our results to the case where the feasible region is expressed by both inequality and equality constraints. This can be done, exploiting the connections between the second order contingent set of the feasible region and the second order derivatives of the constraining functions. By means of theorems of the alternative, we are therefore able to deduce a John type multipliers rule, involving both the Jacobians and the Hessians of the objective and constraining functions. We stress that the multipliers are not fixed but they depend upon the chosen descent direction. In the last section, we analyse some conditions, which guarantee the existence of nonzero multipliers corresponding to the objective functions; following the approach developed by Kawasaki for scalar problems, we consider a constraint qualification, which is weaker than those already introduced, and we show that the Guignard type constraint qualification is useless without convexity assumptions; on the contrary, we introduce a Guignard type condition, which involves also the objective functions and needs no convexity assumptions to achieve the goal.

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