A filter method for nonlinear semidefinite programming with global convergence

In this chapter, an implementation of an interior point filter line-search algorithm for large-scale nonlinear programming proposed by Wachter and Biegler (2005a, b) is presented. As we know, to allow convergence from poor starting points and to enforce progress to the solution, interior point methods both in trust region and in line-search frameworks with exact penalty merit function have been developed. For example, KNITRO uses the l 1 exact penalty function (Byrd, Hribar, & Nocedal, 1999; Byrd, Gilbert, & Nocedal, 2000). On the other hand, Fletcher and Leyffer (2002) proposed filter methods as an alternative to merit functions, as a tool for global convergence guarantee in algorithms for nonlinear optimization. The idea of filter is that trial points are accepted if they improve the objective function value or improve the constraint violation instead of a combination of these two measures defined by the merit function. Even if the filter methods include different heuristics, they have been adapted to barrier methods in a number of ways. For example, Ulbrich et al. (2004) considered a trust-region filter method and accept the trial step on the basis of the norm of the optimality conditions. Benson et al. (2002a) proposed several heuristics using the idea of filter methods, for which improved efficiency is reported compared to their previous merit function approach. The global convergence of an interior point algorithm with a filter line search was analyzed by Wachter and Biegler (2001). Here, the assumptions made for the analysis of the global convergence are less restrictive than those made for line-search interior point methods for nonlinear programming developed, for example, by El-Bakry et al. (1996), Yamashita (1998), or Tits et al. (2003).

In this paper, we propose a robust sequential quadratic programming (SQP) method for nonlinear programming without using any explicit penalty function and filter. The method embeds the modified QP subproblem proposed by Burke and Han (Math Program 43:277---303, 1989) for the search direction, which overcomes the common difficulty in the traditional SQP methods, namely the inconsistency of the quadratic programming subproblems. A non-monotonic technique is employed further in a framework in which the trial point is accepted whenever there is a sufficient relaxed reduction of the objective function or the constraint violation function. A forcing sequence possibly tending to zero is introduced to control the constraint violation dynamically, which is able to prevent the constraint violation from over-relaxing and plays a crucial role in global convergence and the local fast convergence as well. We prove that the method converges globally without the Mangasarian---Fromovitz constraint qualification (MFCQ). In particular, we show that any feasible limit point that satisfies the relaxed constant positive linear dependence constraint qualification is also a Karush---Kuhn---Tucker point. Under the strict MFCQ and the second order sufficient condition, furthermore, we establish the superlinear convergence. Preliminary numerical results show the efficiency of our method.

Nonlinear semidefinite programming (SDP) problems have received a lot of attentions because of large variety of applications. In this paper, we survey numerical methods for solving nonlinear SDP problems. Three kinds of typical numerical methods are described; augmented Lagrangian methods, sequential SDP methods and primal-dual interior point methods. We describe their typical algorithmic forms and discuss their global and local convergence properties which include rate of convergence.

A framework for proving global convergence for a class of line search filter-type methods for nonlinear programming is presented without assuming that the Jacobian has full rank everywhere. The underlying method is based on the filter concept where trial points are accepted, provided there is a sufficient decrease in the objective function or constraints violation function. The proposed methods solve a sequence of quadratic programming subproblems via line search techniques to induce global convergence. Under mild conditions, we will also show that the algorithm converges two step superlinearly when the iterates are near to the solution.

We analyze the rate of local convergence of the augmented Lagrangian method in nonlinear semidefinite optimization. The presence of the positive semidefinite cone constraint requires extensive tools such as the singular value decomposition of matrices, an implicit function theorem for semismooth functions, and variational analysis on the projection operator in the symmetric matrix space. Without requiring strict complementarity, we prove that, under the constraint nondegeneracy condition and the strong second order sufficient condition, the rate of convergence is linear and the ratio constant is proportional to 1/c, where c is the penalty parameter that exceeds a threshold $$\overline{c} > 0$$ .

Duality theories in nonlinear semidefinite programming

In this paper, a formulation for an interior-point Newton method of general nonlinear programming problems is presented. The formulation uses the Coleman-Li scaling matrix. The local convergence and the q-quadratic rate of convergence for the method are established under the standard assumptions of the Newton method for general nonlinear programming.

In this paper, a new algorithm based on SQP method is presented to solve the nonlinear inequality constrained optimization problem. As compared with the other existing SQP methods, per single iteration, the basic feasible descent direction is computed by solving at most two equality constrained quadratic programming. Furthermore, there is no need for any auxiliary problem to obtain the coefficients and update the parameters. Under some suitable conditions, the global and superlinear convergence are shown. Keywords: Global convergence, Inequality constrained optimization, Nonlinear programming problem, SQP method, Superlinear convergence rate.

Direct optimal control methods first discretize a continuous-time Optimal Control Problem (OCP) and then solve the resulting Nonlinear Program (NLP). Sequential Quadratic Programming (SQP) is a popular family of algorithms to solve this finite dimensional optimization problem. In the specific case of a least squares cost, the Generalized Gauss-Newton (GGN) method is a popular approach which works very well under some assumptions. This paper proposes a Sequential Convex Quadratic Programming (SCQP) scheme which exploits additional convexities in the NLP in order to generalize the GGN algorithm, possibly extend its applicability and improve its local convergence. These properties are studied in detail for the proposed SCQP algorithm, which will be compared to the classical GGN method using a numerical case study of the optimal control of an inverted pendulum.

Constrained optimization for microbial fermentation was studied. For optimization, we used not the maximum principle but a nonlinear programming method because of the need to consider many metabolic reactions. In the case of L-lysine fermentation, the optimization problem in L-lysine production was formulated as a nonlinear programming problem. In general, the state equations based on material balances are represented as differential equations, but such equations which are dependent on time can not be applied to a nonlinear programming problem. Therefore, the state equations were made discrete in a time base, and a new single vector which is not dependent on time was substituted. From these formulae, the objective function and the constraints using nonlinear programming problem were defined as the amount of L-lysine produced, and as a metabolic reaction model and empirical equations, respectively. Computer program was developed to solve this constrained nonlinear programming problem. The applied algorithm of the computer programming was a sequential quadratic programming method (SQP method). When the constrained nonlinear programming problem is solved using the SQP method, the maximum amount of L-lysine produced and the optimal feeding rate of L-threonine could be calculated. From the calculated results, it was clear that introduction of the equality and inequality constraints was easy. L-Lysine at a concentration up to 75.3 g/l could be produced when the fermentation was carried out under optimal conditions.

Global and local convergence of a penalty-free method for nonlinear programming

The aim of this paper is to develop a nonlinear programming method for computing the elements of the interval-valued cores of n-person cooperative games in which coalitions’ values are expressed with intervals, which are often called interval-valued n-person cooperative games for short. With finding out the maximum satisfactory degree in the situation with the features of inclusion and/or overlap relations between intervals, this paper tries to explore the cooperation chance in this type of cooperative games. Firstly, we define the concept of interval-valued cores of interval-valued n-person cooperative games and satisfactory degrees (or ranking indexes) of comparing intervals with the features of inclusion and/or overlap relations. Hereby, we propose the auxiliary nonlinear programming model and method for solving interval-valued cores of any interval-valued n-person cooperative games. The developed method can provide cooperative chance under the situation of inclusion and/or overlap relations between intervals, while the traditional interval ranking method may not assure that the interval-valued cores exist. This method is a complement to the traditional methods rather than the alternative one. The feasibility and applicability of the model and method proposed in this paper are illustrated with a numerical example.

Reduced order output feedback control design for PDE systems using proper orthogonal decomposition and nonlinear semidefinite programming

In this paper, a modified Liu–Storey conjugate gradient method is proposed. The method can generate sufficient descent directions for non-linear unconstrained optimization problems. A global convergence result is established when the line search fulfils the strong Wolfe conditions. Moreover, the -linear convergence rate of the methods is proved. The extensive numerical results show that the proposed method is efficient for the unconstrained optimization problems in the CUTEr library.

Recent applications of optimization methods in building frame design practice include optimality criteria method [1,2,3] and the conventional LP method. To use the recurrence relationship in the optimality criteria method, the Lagrange multipliers associated with each active constraint, and the gradients of the weight function and the active constraints must first be determined. But there derivations are lengthy and tedius. Therefore, we try to use the ‘double secant method for nonlinear mathematical programming ‘developed by Tsung-Wu Lin [4] for elastic optimal design. The major advantage of the latest method is that it provides double secants to make the search of correct direction quickly with smaller maximum error at each iterate. Nevertheless, there are still too many iterations required to reach an optimum design if the double secant method alone is used. It can become more effective if we determine first the initial design variables, or the initial member sizes, by applying the stress-ratio method and the shifting method alternatively. The amount of calculation can also be reduced by applying the reanalysis technique or Gauss-Seidal iteration method in the process of nonlinear mathematical programming. Numerical examples have shown that after the initial design variables were determined after two or more iterations, only several iterations were needed for the double secant method.