The Space Of Star-Shaped Sets And Its Applications In Optimization Problems

There are a large number of optimization problems expressed by non-differentiable functions. Directionally differentiable functions are an important class of non-differentiable functions, and attention has been focused on differential properties of this class of non-differentiable functions. This dissertation first establishes the theory of the space of star-shaped sets, based on which a new class of non-differentiable functions-star differentiable functions, is introduced, the differential properties of the star differentiable functions are discussed, and optimality conditions for star differentiable optimization problems are studied.The main results obtained in this dissertation may be summarized as follows:1. Chapter2proposes the notion of the space of star-shaped sets and explores the properties of this space. The partial order (?), inverse sum operation (?) and inverse scalar multiplication⊙are introduced, satisfying the cancelation law, so that the space of star-shaped sets becomes a normed Riesz space and topological vector lattice. It is proved that the space of non-negative positive homogeneous continuous functions is isomorphic to the space of star-shaped sets. The corresponding partial order(?), inverse sum operation (?) and inverse scalar multiplication⊙are obtained and they also obey the cancelation law from which the space of non-negative positive homogeneous continuous functions is a normed Riesz space and topological vector lattice.2. Chapter3, based on the theory for the space of star-shaped sets, defines a new class of functions in the set of directionally differentiable functions-star-shaped differen-tiable functions, whose directional derivatives can be expressed as the differences between positively homogenous continuous functions. Its differential is names as the star-shaped differential, which is a pair of star-shaped set. We give an example to il-lustrate that the space of star-shaped differentiable functions is bigger than the space of quasi-differentiable functions in the sense of Demyanov and Rubinov (1986), and we establish the four basic arithmetic operations, the formulas of star-differentials for composite operation and pointwise maximum and minimum operations, as well as the mean-value theorem for the star-shaped differentiable functions.3. Chapter4discusses optimality conditions for optimization problems of the star-shaped differentiable functions. First of all, the optimality conditions for a un-constrained star-shaped optimization problem, including necessary conditions and sufficient conditions, are characterized by use of star-shaped differentials of the objective function. Secondly, optimality conditions for the inequality constrained problem of star-shaped functions are studied, including the necessary conditions characterized by star-shaped differentials, the necessary conditions based on the notion of regularity for the constrained set, as well as the Fritz-John necessary conditions for an local minimizer. Finally, the necessary optimality conditions for equality constrained star-shaped optimization problems are derived by using the penalty function method.