Solving nonlinear programming problems with very many constraints

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.

In this paper, a new algorithm, namely, a reduced Hessian sequential quadratic programming (SQP) method, for solving electromagnetic inverse problems is proposed. The electromagnetic inverse problem is considered to be a constrained nonlinear programming. The reduced Hessian SQP method finds the solution of this constrained nonlinear programming by solving a sequential of quadratic programming subproblems. The reduced Hessian scheme is applied to reduce the requirement of computational memory of the basic SQP method for large inverse problems. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed method, and some comparisons show that the proposed method has a better convergence and a faster speed than the previous methods.

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.

Dynamics analysis and time-optimal motion planning for unmanned quadrotor transportation systems

Future wireless systems must be designed to meet high data rates and a variety of services such as voice, video, etc. which consume more energy. Therefore, designing energy-efficient systems in recent years has become an inevitable trend. In this paper energy efficient resource allocation in orthogonal frequency division multiple access (OFDMA) systems is discussed. After formulating the resource allocation problem, maximizing-energy efficiency (EE)-lower-bound-based downlink subcarrier assignment (MDSA) algorithm is used to solve the problem that allocates subcarrier and power recursively. In this algorithm, first primary subcarriers are assigned to users and then power is allocated and this process continues recursively until all subcarriers are assigned to the users. Power allocation is done by the use of proposed sequential Quadratic programming (SQP) method. Then, simulation results are compared with bisection based power adaptation (BPA) algorithm method which has different power allocation. Results show that power allocation by SQP method has higher energy efficiency. Moreover, SQP shows higher convergence rate in solving the problem (finding the answer) in comparison to BPA method with less simulation time.

Sequential quadratic programming (SQP) methods form a class of highly efficient algorithms for solving nonlinearly constrained optimization problems. Although second derivative information may often be calculated, there is little practical theory that justifies exact-Hessian SQP methods. In particular, the resulting quadratic programming (QP) subproblems are often nonconvex, and thus finding their global solutions may be computationally nonviable. This paper presents a second derivative SQP method based on quadratic subproblems that are either convex, and thus may be solved efficiently, or need not be solved globally. Additionally, an explicit descent-constraint is imposed on certain QP subproblems, which “guides” the iterates through areas in which nonconvexity is a concern. Global convergence of the resulting algorithm is established.

In this paper, the design of non-uniform concentric circular antenna arrays (CCAAs) of isotropic radiators with optimum side lobe level reduction is presented. The newly-proposed global evolutionary optimization methods; namely: the biogeography based optimization (BBO) is used to determine a set of weights that provides a radiation pattern with optimum side lobe level reduction with the constraint of a fixed major lobe beamwidth. The results obtained from BBO are compared with those obtained using the Matlab function Fmincon which uses a sequential quadratic programming (SQP) method. The comparison shows that the design of non-uniformly excited CCAAs using SQP method provides a side lobe level reduction that is better than that obtained using global stochastic optimization methods, indicating that global optimization techniques are not really needed in this problem.

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.

Sequential Quadratic Programming (SQP) denotes an established class of methods for solving nonlinear optimization problems via an iterative sequence of Quadratic Programs (QPs). Several approaches in the literature have established local and global convergence of the method. This paper considers a variant to SQP that, instead of solving each QP, at each iteration follows one gradient step along a direction that is proven to be a descent direction for an augmented Lagrangian of the nonlinear problem, and in turn is used to generate the next QP. We prove global convergence to a critical point of the original nonlinear problem via a line search that requires the same assumptions as the SQP method. The method is then applied to nonlinear Model Predictive Control (MPC). To simplify the problem formulation and achieve faster convergence, we propose a locally convergent reformulation. An important speed-up is observed in practice via a specific initialization. The computational efficiency of the proposed method is finally shown in a numerical example.

Optimizations of a photoresist coating process for photolithography in wafer manufacture via a radial basis neural network: A case study

This study analyzed contour distortions, wear and tensile properties of polypropylene (PP) components applied in the interior coffer of automobiles. A hybrid method integrating a trained generalized regression neural network (GRNN) and a sequential quadratic programming (SQP) method is proposed to determine an optimal parameter setting of the injection‐molding process. The specimens were prepared under different injection‐molding conditions by changing melting temperatures, injection speeds, and injection pressures. Average contour distortions at six critical locations, wear and tensile properties were selected as the quality targets. Sixteen experimental runs, based on a Taguchi orthogonal array table, were utilized to train the GRNN and then the SQP method was applied to search for an optimal setting. The trained GRNN was capable of predicting average contour distortions, wear and tensile properties at various injection‐molding conditions. In addition, the analysis of variance (ANOVA) was implemented to identify significant factors for the molding process and the proposed algorithm was compared with traditional schemes like the Taguchi method and the design of experiments (DOE) approach. Copyright © 2007 John Wiley & Sons, Ltd.

Optimal feed control for the fed-batch fermentation process of ethanol production is studied. Additional inequality constraints are introduced in this optimization problem to assure the optimal solution in a reality region. Introducing an updating rule of augmented Lagrange multipliers to handle these inequality constraints, iterative dynamic programming can be used in a straightforward manner for the optimization of fed-batch fermentors. To obtain more accurate solution a method of sequential quadratic programming can be used to solve this problem again. As a result of this optimal control, the maximum production at final time is very close to the theoretical yield. Although sequential quadratic programming can be rapid convergence to the optimal solution, but very good initial starting points has to be used to ensure obtaining the global optimum. Experimental works were used to validate this study. The simulated results could fit the experiments satisfactorily.

This paper investigates the benefit of using a 3D road geometry based optimal powertrain control strategy in reducing the fuel consumption of heavy trucks. The optimal control, which applies a sequential quadratic programming (SQP) method, is designed to predict the optimal truck velocity trajectory, based on the road geometry with the consideration of fuel consumption and travel time. The fuel consumption baseline is developed based on an engineering drive cycle. Computer simulations of a Class 8 truck are conducted with Intermap real 3D road geometry. Simulation results show that the optimal control strategy is able to reduce the fuel consumption with equal or even shorter travel time, when compared to the defined baseline.

We analyze sequential quadratic programming (SQP) methods to solve nonlinear constrained optimization problems that are more flexible in their definition than standard SQP methods. The type of flexibility introduced is motivated by the necessity to deviate from the standard approach when solving large problems. Specifically we no longer require a minimizes of the QP subproblem to be determined or particular Lagrange multiplier estimates to be used. Our main focus is on an SQP algorithm that uses a particular augmented Lagrangian merit function. New results are derived for this algorithm under weaker conditions than previously assumed; in particular, it is not assumed that the iterates lie on a compact set.

Comparison of SQP and AL algorithms for deterministic constrained production optimization of hydrocarbon reservoirs