Xiaojiao Tong

A smoothing method for solving portfolio optimization with CVaR and applications in allocation of generation asset

This paper focuses on the computation issue of portfolio optimization with scenario-based CVaR. According to the semismoothness of the studied models, a smoothing technology is considered, and a smoothing SQP algorithm then is presented. The global convergence of the algorithm is established. Numerical examples arising from the allocation of generation assets in power markets are done. The computation efficiency between the proposed method and the linear programming (LP) method is compared. Numerical results show that the performance of the new approach is very good. The remarkable characteristic of the new method is threefold. First, the dimension of smoothing models for portfolio optimization with scenario-based CVaR is low and is independent of the number of samples. Second, the smoothing models retain the convexity of original portfolio optimization problems. Third, the complicated smoothing model that maximizes the profit under the CVaR constraint can be reduced to an ordinary optimization model equivalently. All of these show the advantage of the new method to improve the computation efficiency for solving portfolio optimization problems with CVaR measure.

A smoothing projected Newton-type algorithm for semi-infinite programming

Abstract This paper presents a smoothing projected Newton-type method for solving the semi-infinite programming (SIP) problem. We first reformulate the KKT system of the SIP problem into a system of constrained nonsmooth equations. Then we solve this system by a smoothing projected Newton-type algorithm. At each iteration only a system of linear equations needs to be solved. The feasibility is ensured via the aggregated constraint under some conditions. Global and local superlinear convergence of this method is established under some standard assumptions. Preliminary numerical results are reported.

The Lagrangian Globalization Method for Nonsmooth Constrained Equations

The difficulty suffered in optimization-based algorithms for the solution of nonlinear equations lies in that the traditional methods for solving the optimization problem have been mainly concerned with finding a stationary point or a local minimizer of the underlying optimization problem, which is not necessarily a solution of the equations. One method to overcome this difficulty is the Lagrangian globalization (LG for simplicity) method. This paper extends the LG method to nonsmooth equations with bound constraints. The absolute system of equations is introduced. A so-called Projected Generalized-Gradient Direction (PGGD) is constructed and proved to be a descent direction of the reformulated nonsmooth optimization problem. This projected approach keeps the feasibility of the iterates. The convergence of the new algorithm is established by specializing the PGGD. Numerical tests are given.

Worst-case CVaR based portfolio optimization models with applications to scenario planning

This article studies three robust portfolio optimization models under partially known distributions. The proposed models are composed of min–max optimization problems under the worst-case conditional value-at-risk consideration. By using the duality theory, the models are reduced to simple mathematical programming problems where the underlying random variables have a mixture distribution or a box discrete distribution. They become linear programming problems when the loss function is linear. The solutions between the original problems and the reduced ones are proved to be identical. Furthermore, for the mixture distribution, it is shown that the three profit-risk optimization models have the same efficient frontier. The reformulated linear program shows the usability of the method. As an illustration, the robust models are applied to allocations of generation assets in power markets. Numerical simulations confirm the theoretical analysis.

A decoupled semismooth Newton method for optimal power flow

In the deregulated environment, optimization tools that coordinate both system and security and economy are widely used to support system operation and decision. In this paper, we present a new decoupled approach to solve optimal power flow (OPF) and available transfer capability (ATC) problems. First, the KKT system of original optimization problem is reformulated equivalently to nonsmooth equations using a so-called nonlinear complementarity problem (NCP) function. Based on the new reformulation, we then apply the inherent weak-coupling characteristics of power systems and design a decoupled Newton algorithm in which a decomposition-correction strategy is considered. The combination of semismooth Newton method and decoupled strategy shares those advantages of two methods such as fast convergence and saving computation cost. Meanwhile, the new method is supported by theoretical convergence. Numerical examples of both OPF and ATC problems demonstrate that the new algorithm has potential to solve large-scale optimization problems

Available Transfer Capability Calculation Using a Smoothing Pointwise Maximum Function

The determination of the available transfer capability (ATC) is formulated as a problem of finding the solution of a system of equations using the pointwise maximum function, which collapses all operating constraints of the system into one equation. The resulting equations are semismooth. The semismooth equations are solved by the use of a smoothing function. Two solution algorithms are presented. One is a smoothing Newton method in which the smoothing parameter is treated as an independent variable. Another one is a smoothing decoupled Newton method, which incorporates the inherent weak-coupling characteristics of power systems into the algorithm and is suitable for solving large scale problems. The convergence of the two algorithms is studied in detail. A numerical example is presented to illustrate the effectiveness of the proposed methods.

Robust reward–risk ratio optimization with application in allocation of generation asset

In this article, we study reward–risk ratio models under partially known message of random variables, which is called robust (worst-case) performance ratio problem. Based on the positive homogenous and concave/convex measures of reward and risk, respectively, the new robust ratio model is reduced equivalently to convex optimization problems with a min–max optimization framework. Under some specially partial distribution situation, the convex optimization problem is converted into simple framework involving the expectation reward measure and conditional value-at-risk measure. Compared with the existing reward–risk portfolio research, the proposed ratio model has two characteristics. First, the addressed problem combines with two different aspects. One is to consider an incomplete information case in real-life uncertainty. The other is to focus on the performance ratio optimization problem, which can realize the best balance between the reward and risk. Second, the complicated optimization model is transferred into a simple convex optimization problem by the optimal dual theorem. This indeed improves the usability of models. The generation asset allocation in power systems is presented to validate the new models.

A smoothing SQP method for nonlinear programs with stability constraints arising from power systems

This paper investigates a new class of optimization problems arising from power systems, known as nonlinear programs with stability constraints (NPSC), which is an extension of ordinary nonlinear programs. Since the stability constraint is described generally by eigenvalues or norm of Jacobian matrices of systems, this results in the semismooth property of NPSC problems. The optimal conditions of both NPSC and its smoothing problem are studied. A smoothing SQP algorithm is proposed for solving such optimization problem. The global convergence of algorithm is established. A numerical example from optimal power flow (OPF) is done. The computational results show efficiency of the new model and algorithm.

Semi-infinite programming method for optimal power flow with transient stability and variable clearing time of faults

This paper presents a new nonlinear programming problem arising in the control of power systems, called optimal power flow with transient stability constraint and variable clearing time of faults and abbreviated as OTS-VT. The OTS-VT model is converted into a implicit generalized semi-infinite programming (GSIP) problem. According to the special box structure of the reformulated GSIP, a solution method based on bi-level optimization is proposed. The research in this paper has two contributions. Firstly, it generalizes the OTS study to general optimal power flow with transient stability problems. From the viewpoint of practical applications, the proposed research can improve the decision-making ability in power system operations. Secondly, the reformulation of OTS-VT also provides a new background and a type of GSIP in the research of mathematical problems. Numerical results for two chosen power systems show that the methodology presented in this paper is effective and promising.

An iterative method for solving semismooth equations

In this paper, we combine trust region technique with line search technique to develop an iterative method for solving semismooth equations. At each iteration, a trust region subproblem is solved. The solution of the trust region subproblem provides a descent direction for the norm of a smoothing function. By using a backtracking line search, a steplength is determined. The proposed method shares advantages of trust region methods and line search methods. Under appropriate conditions, the proposed method is proved to be globally and superlinearly convergent. In particular, we show that after finitely many iterations, the unit step is always accepted and the method reduces to a smoothing Newton method.