Jinbao Jian

An improved SQP algorithm for solving minimax problems

In this work, an improved SQP method is proposed for solving minimax problems, and a new method with small computational cost is proposed to avoid the Maratos effect. In addition, its global and superlinear convergence are obtained under some suitable conditions.

Outer Approximation and Outer-Inner Approximation Approaches for Unit Commitment Problem

This paper proposes a new separable model for the unit commitment (UC) problem and three deterministic global optimization methods for it ensuring convergence to the global optimum within a desired tolerance. By decomposing a multivariate function into several univariate functions, a tighter outer approximation methodology that can be used to improve the outer approximations of several classical convex programming techniques is presented. Based on the idea of the outer approximation (OA) method and the proposed separable model, an outer-inner approximation (OIA) approach is also presented to solve this new formulation of UC problem. In this OIA approach, the UC problem is decomposed into a tighter outer approximation subproblem and an inner approximation subproblem, where the former leads to a better lower bound than the OA method, and the later provides a better upper bound. The simulation results for systems of up to 100 units with 24 h are compared with those of previously published methods, which show that the OIA approach is very promising due to the excellent performance. The proposed approaches are also applied to the large-scale systems of up to 1000 units with 24 h, and systems of up to 100 units with 96 h and 168 h.

A new superlinearly convergent norm-relaxed method of strongly sub-feasible direction for inequality constrained optimization ☆

Method of feasible directions (MFD) is an important method for solving nonlinearly constrained optimization. However, various types of MFD all need an initial feasible point, which can not be found easily in generally. In addition, the computational cost of some MFD with superlinearly convergent property is rather high. On the other hand, the strongly sub-feasible direction method does not need an initial feasible point, but most of the proposed algorithm do not have the superlinearly convergent property, and can not guarantee that the iteration point is feasible after finite iterations. In this paper, we present a new superlinearly convergent algorithm with arbitrary initial point. At each iteration, a master direction is obtained by solving one direction finding subproblem (DFS), and an auxiliary direction is yielded by an explicit formula. After finite iterations, the iteration point goes into the feasible set and the master direction is a feasible direction of descent. Since a new generalized projection technique is contained in the auxiliary direction formula, under some mild assumptions without the strict complementarity, the global convergence and superlinear convergence of the algorithm can be obtained.

A new norm-relaxed method of strongly sub-feasible direction for inequality constrained optimization

Combining the norm-relaxed method of feasible direction (MFD) with the idea of strongly sub-feasible direction method, we present a new convergent algorithm with arbitrary initial point for inequality constrained optimization. At each iteration, the new algorithm solves one direction finding subproblem (DFS) which always possesses a solution. Some good properties of the new algorithm are that it can unify automatically the operations of initialization (Phase I) and optimization (Phase II) and the number of the functions satisfying the inequality constrains is nondecreasing, particularly, a feasible direction of descent can be obtained by solving DFS whenever the iteration point gets into the feasible set. Under some mild assumptions without the linear independence, the global and strong convergence of the algorithm can be obtained.

An improved SQP algorithm for inequality constrained optimization

Abstract.In this paper, the feasible type SQP method is improved. A new algorithm is proposed to solve nonlinear inequality constrained problem, in which a new modified method is presented to decrease the computational complexity. It is required to solve only one QP subproblem with only a subset of the constraints estimated as active per single iteration. Moreover, a direction is generated to avoid the Maratos effect by solving a system of linear equations. The theoretical analysis shows that the algorithm has global and superlinear convergence under some suitable conditions. In the end, numerical experiments are given to show that the method in this paper is effective.

Global Optimization of Non-Convex Hydro-Thermal Coordination Based on Semidefinite Programming

In order to reduce the fuel cost of thermal units, a semidefinite programming (SDP) method is used to solve a hydrothermal coordination (HTC) optimization problem. By manipulating the structure of decision variable matrix, the original nonconvex problem is reformulated into a convex SDP relaxation model without sacrificing the nonlinear relation among hydropower generation, reservoir storage volume, and discharge water. A global minimum is therefore guaranteed and well-developed convex optimization theories can thus be employed to solve the problem. Both sparse matrix techniques and a simplified SDP model are discussed to reduce computational cost. One mostly used HTC case is employed to test the performance of the proposed method. Detailed comparisons between the proposed and other methods show that the final result of SDP model is by far the best result ever. In addition, a large-sized HTC case shows the potential of SDP in practical use. Finally, we prove that as a relaxation technique, the SDP solution, which satisfies all the constraints, is indeed the optimal solution of the original nonconvex problem.

A Superlinearly Convergent Implicit Smooth SQP Algorithm for Mathematical Programs with Nonlinear Complementarity Constraints

This paper discusses a special class of mathematical programs with nonlinear complementarity constraints, its goal is to present a globally and superlinearly convergent algorithm for the discussed problems. We first reformulate the complementarity constraints as a standard nonlinear equality and inequality constraints by making use of a class of generalized smoothing complementarity functions, then present a new SQP algorithm for the discussed problems. At each iteration, with the help of a pivoting operation, a master search direction is yielded by solving a quadratic program, and a correction search direction for avoiding the Maratos effect is generated by an explicit formula. Under suitable assumptions, without the strict complementarity on the upper-level inequality constraints, the proposed algorithm converges globally to a B-stationary point of the problems, and its convergence rate is superlinear.

Generalized monotone line search SQP algorithm for constrained minimax problems

In this article, non-linear minimax problems with general constraints are discussed. By means of solving one quadratic programming an improved direction is yielded and a second-order correction direction can also be at hand via one system of linear equations. So a new algorithm for solving the discussed problems is presented. In connection with a special merit function, the generalized monotone line search is used to yield the step size at each iteration. Under mild conditions, we can ensure global and superlinear convergence. Finally, some numerical experiments are operated to test our algorithm, and the results demonstrate that it is promising.

Tight Relaxation Method for Unit Commitment Problem Using Reformulation and Lift-and-Project

This paper presents a novel method to solve the unit commitment (UC) problem by solving a sequence of increasingly tight continuous relaxations based on the techniques of reformulation and lift-and-project (L&P). After projecting the power output of unit onto [0, 1], the continuous relaxation of the UC problem can be tightened with the reformulation techniques. Then, a tighter model which is called L&P-TMIP is established by strengthening the continuous relaxation of the feasible region of the tight UC problem iteratively using L&P. High-quality suboptimal solutions can be obtained from the solutions of the relaxation for this tight model. The simulation results for realistic instances that range in size from 10 to 200 units over a scheduling period of 24 h show that the proposed tight relaxation method is competitive with general-purpose mixed integer programming solvers based methods for large-scale UC problems due to the excellent performance and the good quality of the solutions it generates.

A deterministic method for the unit commitment problem in power systems

The unit commitment (UC) problem in power systems is generally formulated as a large-scale nonlinear mixed-integer combinatorial optimization problem, which is difficult to solve. This paper presents a deterministic method named cut-and-branch for solving UC, which is based on cuts and branch-and-bound search as well as heuristic rounding technique. First, a suitable mixed-integer quadratic programming (MIQP) model of UC is presented by some linearization technique, then the MIQP is solved by the proposed cut-and-branch method. In the proposed method, two classes of cuts are introduced to give a stronger representation of the corresponding continuous relaxed problem: one is the approximate integer cut derived from a natural understanding of the problem which is simple but highly efficient, and the other is the generalized flow cover inequality. Furthermore, the continuous relaxed problem incorporating the proposed cuts can obtain some better initial feasible solutions and reduce the numbers of nodes during the branch-and-bound search. The simulation results for six systems with up to 100 units and 24h show that the proposed method has nice convergence, which can find global optimal solution in theory.