The sufficient optimality conditions and duality results have recently been given for the following generalised convex programming problem:where the functions / and g satisfyIt is shown here that a relaxation defining the above generalised convexity leads to a new class of multi-objective problems which preserves the sufficient optimality and duality results in the scalar case, and avoids the major difficulty of verifying that the inequality holds for the same function rj (., .). Further, this relaxation allows one to treat certain nonlinear multi-objective fractional programming problems and some other classes of nonlinear (composite) problems as special cases.

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The trust-region problem, which minimizes a nonconvex quadratic function over a ball, is a key subproblem in trust-region methods for solving nonlinear optimization problems. It enjoys many attractive properties such as an exact semi-definite linear programming relaxation (SDP-relaxation) and strong duality. Unfortunately, such properties do not, in general, hold for an extended trustregion problem having extra linear constraints. This paper shows that two useful and powerful features of the classical trust-region problem continue to hold for an extended trust-region problem with linear inequality constraints under a new dimension condition. First, we establish that the class of extended trust-region problems has an exact SDP-relaxation, which holds without the Slater constraint qualification. This is achieved by proving that a system of quadratic and affine functions involved in the model satisfies a range-convexity whenever the dimension condition is fulfilled. Second, we show that the dimension condition together with the Slater condition ensures that a set of combined first and second-order Lagrange multiplier conditions is necessary and sufficient for global optimality of the extended trust-region problem and consequently for strong duality. Through simple examples we also provide an insightful account of our development from SDP-relaxation to strong duality. Finally, we show that the dimension condition is easily satisfied for the extended trust-region model that arises from the reformulation of a robust least squares problem (LSP) as well as a robust second order cone programming model problem (SOCP) as an equivalent semi-definite linear programming problem. This leads us to conclude that, under mild assumptions, solving a robust (LSP) or (SOCP) under matrix-norm uncertainty or polyhedral uncertainty is equivalent to solving a semi-definite linear programming problem and so, their solutions can be validated in polynomial time. * The authors are grateful to the referees for their valuable suggestions and helpful comments which have contributed to the final preparation of the paper.

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|A notion of approximate Jacobian matrix is introduced for a continuous vector-valued map. It is shown for instance that the Clarke generalized Jacobian is an approximate Jacobian for a locally Lipschitz map. The approach is based on the idea of convexi cators of real-valued functions. Mean value conditions for continuous vector-valued maps and Taylor's expansions for continuously Gâteaux di erentiable functions (i.e. C 1-functions) are presented in terms of approximate Jacobians and approximate Hessians respectively. Second-order necessary, and su cient conditions for optimality and convexity of C 1-functions are also given.

A class of functions, called pre-invex, is defined. These functions are more general than convex functions and when differentiable are invex. Optimality conditions and duality theorems are given for both scalar-valued and vector-valued programs involving pre-invex functions.