Some Properties And Applications Of Conic Optimization And Its Dual Problem

Conic optimization (CO) is a particular case of convex programming, and it is also an extension of linear programming.In conic optimization one minimizes (or maximizes) a linear objective function over the intersection of an affine space with a regular cone in finite dimension.This problem comprises of linear programming(LP), convex quadratic programming(QCQP),semidefinite programming(SDP), second-order conic optimization(SOCP).From its model,similar to the linear programming,it is known that the constraint qualification is not only nonlinearly but also convex. In recent years, conic optimization has been one of the most active research areas in mathematical programming because its theory and algotithms have developed greatly and its numerous applications have been found in portfolio optimization, minimum risk arbitrage, and approximating covariance matrices,and so on.On conic optimization, we work over some properties of cone and its dual cone, and solve the relations between special cones (obtuse cone, orthogonal cone, superior and inferior obtuse cones) and their dual cones.Furthermore, sufficient and necessary condition under which they exist are proved detailed . Based on much knowledge, contrasting to linear programming; we extend duality theorem (including weak duality theorem and strong duality theorem), complementary slack theorem to conic optimization. Hence we find out some significative conclusions and existing conditions under which their duality gap is zero of two optimizations.This article is mostly made up of theory study and application practice. In first part it pays attention to theory study. Afterwards, based on linear programming, we introduce some properties of conic optimization and its dual problem, which are a kind of especial convex programming. In second part, we apply these theories to practice, such as least-squares problems, polynomial solution and approximating covariance matrices, and other aspects .All of these applications show that study on conic optimization is necessary. For detail, we conclude them as follows:The models, foreign and internal study situations on conic optimization and its dual problem are introduced in Chapter one, and we point out the key problem and research matter.Chapter two introduces some primary properties of a cone and its dual cone. Furthermore, some properties, sufficient and necessary condition of special cones and their dual cones are analyzed and proved detailed.Chapter three is the most principal part. In third chapter, some important properties of conic optimization and its dual problem are detailed introduced. We utilize Nesterov and Todd's homogeneous model to estimate the existence of solutions between conic optimization and its dual conic optimization. Similar to linear programming, we give the "KKT" condition of the two optimizations. Through the universal duality in conic convex optimization, we gain the existence condition under which the duality gap of the two programmings is zero. Accordingly, we know the connection of the universal duality and boundness of a primal-dual pair feasible set.We apply these theories to practical applications in chapter four.In chapter five we make many conclusions and bring forward research expectation for the future.conic optimization; dual conic optimization; duality gap; universal duality...