The Space of StarShaped Sets and Its Applications in Optimization Problems 

Author  PanShaoRong 
Tutor  ZhangLiWei 
School  Dalian University of Technology 
Course  Operational Research and Cybernetics 
Keywords  nondifferentiable optimization starshaped set normed Riesz space Banach Kspace stardifferential meanvalue theorem optimality FritzJohn condition 
CLC  O224 
Type  PhD thesis 
Year  2013 
Downloads  16 
Quotes  0 
There are a large number of optimization problems expressed by nondifferentiable functions. Directionally differentiable functions are an important class of nondifferentiable functions, and attention has been focused on differential properties of this class of nondifferentiable functions. This dissertation first establishes the theory of the space of starshaped sets, based on which a new class of nondifferentiable functionsstar 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 starshaped 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 starshaped sets becomes a normed Riesz space and topological vector lattice. It is proved that the space of nonnegative positive homogeneous continuous functions is isomorphic to the space of starshaped 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 nonnegative positive homogeneous continuous functions is a normed Riesz space and topological vector lattice.2. Chapter3, based on the theory for the space of starshaped sets, defines a new class of functions in the set of directionally differentiable functionsstarshaped differentiable functions, whose directional derivatives can be expressed as the differences between positively homogenous continuous functions. Its differential is names as the starshaped differential, which is a pair of starshaped set. We give an example to illustrate that the space of starshaped differentiable functions is bigger than the space of quasidifferentiable functions in the sense of Demyanov and Rubinov (1986), and we establish the four basic arithmetic operations, the formulas of stardifferentials for composite operation and pointwise maximum and minimum operations, as well as the meanvalue theorem for the starshaped differentiable functions.3. Chapter4discusses optimality conditions for optimization problems of the starshaped differentiable functions. First of all, the optimality conditions for a unconstrained starshaped optimization problem, including necessary conditions and sufficient conditions, are characterized by use of starshaped differentials of the objective function. Secondly, optimality conditions for the inequality constrained problem of starshaped functions are studied, including the necessary conditions characterized by starshaped differentials, the necessary conditions based on the notion of regularity for the constrained set, as well as the FritzJohn necessary conditions for an local minimizer. Finally, the necessary optimality conditions for equality constrained starshaped optimization problems are derived by using the penalty function method.