A recursively feasible and convergent Sequential Convex Programming procedure to solve non-convex problems with linear equality constraints

A computationally efficient method to solve non-convex programming problems with linear equality constraints is presented. The proposed method is based on a recursively feasible and descending sequential convex programming procedure proven to converge to a locally optimal solution. Assuming that the first convex problem in the sequence is feasible, these properties are obtained by convexifying the non-convex cost and inequality constraints with inner-convex approximations. Additionally, a computationally efficient method is introduced to obtain inner-convex approximations based on Taylor series expansions. These Taylor-based inner-convex approximations provide the overall algorithm with a quadratic rate of convergence. The proposed method is capable of solving problems of practical interest in real-time. This is illustrated with a numerical simulation of an aerial vehicle trajectory optimization problem on commercial-of-the-shelf embedded computers

A Partially Feasible Distributed SQO Method for Two-block General Linearly Constrained Smooth Optimization

This paper discusses a class of two-block smooth large-scale optimization problems with both linear equality and linear inequality constraints, which have a wide range of applications, such as economic power dispatch, data mining, signal processing, etc.Our goal is to develop a novel partially feasible distributed (PFD) sequential quadratic optimization (SQO) method (PFD-SQO method) for this kind of problems. The design of the method is based on the ideas of SQO method and augmented Lagrangian Jacobian splitting scheme as well as feasible direction method,which decomposes the quadratic optimization (QO) subproblem into two small-scale QOs that can be solved independently and parallelly. A novel disturbance contraction term that can be suitably adjusted is introduced into the inequality constraints so that the feasible step size along the search direction can be increased to 1. The new iteration points are generated by the Armijo line search and the partially augmented Lagrangian function that only contains equality constraints as the merit function. The iteration points always satisfy all the inequality constraints of the problem. The theoretical properties, such as global convergence, iterative complexity, superlinear and quadratic rates of convergence of the proposed PFD-SQO method are analyzed under appropriate assumptions, respectively. Finally, the numerical effectiveness of the method is tested on a class of academic examples and an economic power dispatch problem, which shows that the proposed method is quite promising

An Alternating Trust Region Algorithm for Distributed Linearly Constrained Nonlinear Programs, Application to the AC Optimal Power Flow

A novel trust region method for solving linearly constrained nonlinear programs is presented. The proposed technique is amenable to a distributed implementation, as its salient ingredient is an alternating projected gradient sweep in place of the Cauchy point computation. It is proven that the algorithm yields a sequence that globally converges to a critical point. As a result of some changes to the standard trust region method, namely a proximal regularisation of the trust region subproblem, it is shown that the local convergence rate is linear with an arbitrarily small ratio. Thus, convergence is locally almost superlinear, under standard regularity assumptions. The proposed method is successfully applied to compute local solutions to alternating current optimal power flow problems in transmission and distribution networks. Moreover, the new mechanism for computing a Cauchy point compares favourably against the standard projected search as for its activity detection properties

A second derivative SQP method: local convergence

In [19], we gave global convergence results for a second-derivative SQP method for minimizing the exact ℓ1-merit function for a fixed value of the penalty parameter. To establish this result, we used the properties of the so-called Cauchy step, which was itself computed from the so-called predictor step. In addition, we allowed for the computation of a variety of (optional) SQP steps that were intended to improve the efficiency of the algorithm. \ud \ud Although we established global convergence of the algorithm, we did not discuss certain aspects that are critical when developing software capable of solving general optimization problems. In particular, we must have strategies for updating the penalty parameter and better techniques for defining the positive-definite matrix Bk used in computing the predictor step. In this paper we address both of these issues. We consider two techniques for defining the positive-definite matrix Bk—a simple diagonal approximation and a more sophisticated limited-memory BFGS update. We also analyze a strategy for updating the penalty paramter based on approximately minimizing the ℓ1-penalty function over a sequence of increasing values of the penalty parameter.\ud \ud Algorithms based on exact penalty functions have certain desirable properties. To be practical, however, these algorithms must be guaranteed to avoid the so-called Maratos effect. We show that a nonmonotone varient of our algorithm avoids this phenomenon and, therefore, results in asymptotically superlinear local convergence; this is verified by preliminary numerical results on the Hock and Shittkowski test set

On the Local and Global Convergence of a Reduced Quasi-Newton Method

In optimization in R^n with m nonlinear equality constraints, we study the local convergence of reduced quasi-Newton methods, in which the updated matrix is of order n-m. In particular, we give necessary and sufficient conditions for q-superlinear convergence (in one step). We introduce a device to globalize the local algorithm which consists in determining a step on an arc in order to decrease an exact penalty function. We give conditions so that asymptotically the step will be equal to one

An interior-point method for mpecs based on strictly feasible relaxations.

An interior-point method for solving mathematical programs with equilibrium constraints (MPECs) is proposed. At each iteration of the algorithm, a single primaldual step is computed from each subproblem of a sequence. Each subproblem is defined as a relaxation of the MPEC with a nonempty strictly feasible region. In contrast to previous approaches, the proposed relaxation scheme preserves the nonempty strict feasibility of each subproblem even in the limit. Local and superlinear convergence of the algorithm is proved even with a less restrictive strict complementarity condition than the standard one. Moreover, mechanisms for inducing global convergence in practice are proposed. Numerical results on the MacMPEC test problem set demonstrate the fast-local convergence properties of the algorithm