Nonlinear Programming Algorithm Research Articles

Exact sensitivity analysis of stresses and lightweight design of Timoshenko composite beams

The paper describes the novel optimization techniques for lightweight design of composite beams. The optimization model is to find width and depth of composite beams to minimize the mass of beams under the stiffness, strength and delamination failure constraints. The exact formulae for displacements, stresses and their sensitivities with respect to width and depth are derived using Timoshenko continuous beam theory. The analytical stiffness, strength and delamination failure functions, and their gradients are obtained using the exact expressions of displacements and stresses, and their sensitivities. The mass and its gradient are also expressed analytically. The standard gradient-based nonlinear programming algorithms are employed to solve lightweight design problems of composite beams. The lightweight designs of composite beams are performed using the proposed optimization techniques. Three standard gradient-based optimization methods (SQP, interior-point and active-set) using exact derivatives converge to the same lightweight design. However, gradient-based algorithms using finite difference sensitivities may not lead to optimal lightweight designs. It is necessary to develop exact sensitivity analysis method instead of the difference methods for gradient-based algorithms.

An inertia-free filter line-search algorithm for large-scale nonlinear programming

We present a filter line-search algorithm that does not require inertia information of the linear system. This feature enables the use of a wide range of linear algebra strategies and libraries, wh...

A matheuristic approach to solve the multiobjective beam angle optimization problem in intensity‐modulated radiation therapy

AbstractSelecting a suitable set of beam angles is an important but difficult task in intensity‐modulated radiation therapy (IMRT) for cancer treatment. From a single objective point of view, this problem, known as the beam angle optimization (BAO) problem, is solved by finding a beam angle configuration (BAC) that leads to the best dose distribution, according to some objective function. As there exists a trade‐off between the main goals in IMRT (to irradiate the tumor according to some prescription and to avoid surrounding healthy tissue), it makes sense to solve this problem from a multiobjective (MO) point of view. When doing so, a solution of the BAO problem is no longer a single BAC, but instead a set of BACs that lead to a set of dose distributions that, depending on both dose prescription and physician preferences, can be selected as the preferred treatment. We solve this MO problem using a two‐phase strategy. During the first phase, a deterministic local search algorithm is used for selecting a set of locally optimal BACs, according to a single‐objective function. During this search, an optimal dose distribution for each BAC, with respect to the single‐objective function, is calculated using an exact nonlinear programming algorithm. During the second phase, a set of nondominated points is generated for each promising locally optimal BAC and a dominance analysis among them is performed. The output of the procedure is a set of (approximately) efficient BACs that lead to good dose distributions. To demonstrate the viability of the method, the two‐phase strategy is applied to a prostate case.

A stabilized filter SQP algorithm for nonlinear programming

This paper presents a stabilized filter sequential quadratic programming (SQP) method for the general nonlinear optimization problems. The technique of stabilizing the inner quadratic programmings is an efficient strategy for the degenerate problem and brings the local superlinear convergence, while the integrated filter technique works effectively and guarantees the global convergence. The new algorithm works on both the primal and dual variables and solves the problem within the computational complexity comparable to the classical SQP algorithm. For the convergence, we show that (1) it converges either to a Karush---Kuhn---Tucker point at which the cone-continuity property holds, or to a stationary point in the sense of minimizing the constraint violation, and (2) under some second-order sufficient conditions, it converges locally superlinearly without any constraint qualifications. Our preliminary numerical results on a set of CUTEr test problems as well as on degenerate problems demonstrate the efficiency of the new algorithm.

A Strategy Based on Convex Relaxation for Solving the Oil Refinery Operations Planning Problem

Oil refining is one of the most complex activities in the chemical industry due to its nonlinear nature, which ruins the convexity properties and prevents any guarantees of the global optimality of solutions obtained by local nonlinear programming (NLP) algorithms. Moreover, using global optimization algorithms is often not feasible because they typically require large computational efforts. This paper proposes the use of convex relaxations based on McCormick envelopes for the Refinery Operations Planning Problem (ROPP) that can be used to generate both initial solutions for the ROPP and to estimate optimality gaps for the solutions obtained. The results obtained using data from a real-world refinery suggest that the proposed approach can provide reasonably good solutions for the ROPP, even for cases in which there was no solution available using traditional local NLP algorithms. Furthermore, compared with a global optimization solver, the proposed heuristic is capable of obtaining better solutions in les...

The Multiobjective Trajectory Optimization for Hypersonic Glide Vehicle Based on Normal Boundary Intersection Method

To solve the multiobjective optimization problem on hypersonic glider vehicle trajectory design subjected to complex constraints, this paper proposes a multiobjective trajectory optimization method that combines the boundary intersection method and pseudospectral method. The multiobjective trajectory optimization problem (MTOP) is established based on the analysis of the feature of hypersonic glider vehicle trajectory. The MTOP is translated into a set of general optimization subproblems by using the boundary intersection method and pseudospectral method. The subproblems are solved by nonlinear programming algorithm. In this method, the solution that has been solved is employed as the initial guess for the next subproblem so that the time consumption of the entire multiobjective trajectory optimization problem shortens. The maximal range and minimal peak heat problem is solved by the proposed method. The numerical results demonstrate that the proposed method can obtain the Pareto front of the optimal trajectory, which can provide the reference for the trajectory design of hypersonic glider vehicle.

A Cone-Continuity Constraint Qualification and Algorithmic Consequences

Every local minimizer of a smooth constrained optimization problem satisfies the sequential approximate Karush--Kuhn--Tucker (AKKT) condition. This optimality condition is used to define the stopping criteria of many practical nonlinear programming algorithms. It is natural to ask for conditions on the constraints under which AKKT implies KKT. These conditions will be called strict constraint qualifications (SCQs). In this paper we define a cone-continuity property (CCP) that will be shown to be the weakest possible SCQ. Its relation to other constraint qualifications will also be clarified. In particular, it will be proved that CCP is strictly weaker than the constant positive generator constraint qualification.

Convergence of a Three-Dimensional Dwindling Filter Algorithm Without Feasibility Restoration Phase

ABSTRACTIn this article, we propose a three-dimensional dwindling filter algorithm for general nonlinear programming. The envelope of the three-dimensional dwindling filter becomes thinner and thinner as the step size approaches zero so that the new filter has more flexibility for the acceptance of the trial step size. Moreover, we show that the feasibility restoration phase, which is always used in traditional filter method, is not needed. The modified limited memory Broyden-Fletcher-Goldfarb-Shanno method is employed in the algorithm, and the update matrices are positive definite when the Lagrangian function is a general convex function. Under mild conditions, the global convergence of the new algorithm is analyzed. The primary numerical experiments are reported to show effectiveness of the proposed algorithm.

Packing ellipsoids by nonlinear optimization

In this paper, continuous and differentiable nonlinear programming models and algorithms for packing ellipsoids in the n-dimensional space are introduced. Two different models for the non-overlapping and models for the inclusion of ellipsoids within half-spaces and ellipsoids are presented. By applying a simple multi-start strategy combined with a clever choice of starting guesses and a nonlinear programming local solver, illustrative numerical experiments are presented.

Dynamic optimization for robust path planning of horizontal oil wells

This paper considers the three-dimensional path planning problem for horizontal oil wells. The decision variables in this problem are the curvature, tool-face angle and switching points for each turn segment in the path, and the optimization objective is to minimize the path length and target error. The optimal curvatures, tool-face angles and switching points can be readily determined using existing gradient-based dynamic optimization techniques. However, in a real drilling process, the actual curvatures and tool-face angles will inevitably deviate from the planned optimal values, thus causing an unexpected increase in the target error. This is a critical challenge that must be overcome for successful practical implementation. Accordingly, this paper introduces a sensitivity function that measures the rate of change in the target error with respect to the curvature and tool-face angle of each turn segment. Based on the sensitivity function, we propose a new optimization problem in which the switching points are adjusted to minimize target error sensitivity subject to continuous state inequality constraints arising from engineering specifications, and an additional constraint specifying the maximum allowable increase in the path length from the optimal value. Our main result shows that the sensitivity function can be evaluated by solving a set of auxiliary dynamic systems. By combining this result with the well-known time-scaling transformation, we obtain an equivalent transformed problem that can be solved using standard nonlinear programming algorithms. Finally, the paper concludes with a numerical example involving a practical path planning problem for a Ci-16-Cp146 well.

Genetic Algorithm based Resource Management for Cognitive Mesh Networks with Real-time and Non-real-time Services

Quality-of-service (QoS) provisioning for a cognitive mesh network (CMN) with heterogeneous services has become a challenging area of research in recent days. Considering both real-time (RT) and non-real-time (NRT) traffic in a multihop CMN, [1] studied cross-layer resource management, including joint access control, route selection, and resource allocation. Due to the complexity of the formulated resource allocation problems, which are mixed-integer non-linear programming, a low-complexity yet efficient algorithm was proposed there to approximately solve the formulated optimization problems. In contrast, in this work, we present an application of genetic algorithm (GA) to re-address the hard resource allocation problems studied in [1]. Novel initialization, selection, crossover, and mutation operations are designed such that solutions with enough randomness can be generated and converge with as less number of attempts as possible, thus improving the efficiency of the algorithm effectively. Simulation results show the effectiveness of the newly proposed GA-based algorithm. Furthermore, by comparing the performance of the newly proposed algorithm with the one proposed in [1], more insights have been obtained in terms of the tradeoff among QoS provisioning for RT traffic, throughput maximization for NRT traffic, and time complexity of an algorithm for resource allocation in a multihop network such as CMN.

Models and Methods of Nonlinear Programming in the CAD System of Railways Routing

This article deals with a new CAD system of railways routing. The route of railway as 3D curve is traditionally presented by two flat curves: plan and longitudinal profile. The plan of route is its projection on horizontal plane XOY and its longitudinal profile is dependent Z(s), where s - is the length of plan from initial point. The software of the new system allows us to receive the optimal project line of the longitudinal profile for each variant of the route plan. Thus, it is an opportunity for objective comparison and selection of the best from them. The new algorithm of nonlinear programming was developed for this purpose. This algorithm is based on essential features of the system of constraints on parameters of unknown optimal project line. In the new CAD system, projection of longitudinal profile is provided many times with different source data due to interrelations with other project tasks. For railway haul (25-30 km), optimization of longitudinal profile is a big dimension problem of nonlinear programming with the number of variables and constraints in a few hundreds and more. The standard algorithms of nonlinear programming demand a lot of machine time for solving this task. The new algorithm is more efficient due to using the features of constraints. Keywords: Basis of linear space, null space of a matrix, objective function, reduced antigradient, system of constraints.

Multi-Period Dynamic Optimization for Large-Scale Differential-Algebraic Process Models under Uncertainty

A technique for optimizing large-scale differential-algebraic process models under uncertainty using a parallel embedded model approach is developed in this article. A combined multi-period multiple-shooting discretization scheme is proposed, which creates a significant number of independent numerical integration tasks for each shooting interval over all scenario/period realizations. Each independent integration task is able to be solved in parallel as part of the function evaluations within a gradient-based non-linear programming solver. The focus of this paper is on demonstrating potential computation performance improvement when the embedded differential-algebraic equation model solution of the multi-period discretization is implemented in parallel. We assess our parallel dynamic optimization approach on two case studies; the first is a benchmark literature problem, while the second is a large-scale air separation problem that considers a robust set-point transition under parametric uncertainty. Results indicate that focusing on the speed-up of the embedded model evaluation can significantly decrease the overall computation time; however, as the multi-period formulation grows with increased realizations, the computational burden quickly shifts to the internal computation performed within the non-linear programming algorithm. This highlights the need for further decomposition, structure exploitation and parallelization within the non-linear programming algorithm and is the subject for further investigation.

The extended supporting hyperplane algorithm for convex mixed-integer nonlinear programming

A new deterministic algorithm for solving convex mixed-integer nonlinear programming (MINLP) problems is presented in this paper: The extended supporting hyperplane (ESH) algorithm uses supporting hyperplanes to generate a tight overestimated polyhedral set of the feasible set defined by linear and nonlinear constraints. A sequence of linear or quadratic integer-relaxed subproblems are first solved to rapidly generate a tight linear relaxation of the original MINLP problem. After an initial overestimated set has been obtained the algorithm solves a sequence of mixed-integer linear programming or mixed-integer quadratic programming subproblems and refines the overestimated set by generating more supporting hyperplanes in each iteration. Compared to the extended cutting plane algorithm ESH generates a tighter overestimated set and unlike outer approximation the generation point for the supporting hyperplanes is found by a simple line search procedure. In this paper it is proven that the ESH algorithm converges to a global optimum for convex MINLP problems. The ESH algorithm is implemented as the supporting hyperplane optimization toolkit (SHOT) solver, and an extensive numerical comparison of its performance against other state-of-the-art MINLP solvers is presented.

On the tuning of predictive controllers: Application of generalized Benders decomposition to the ELOC problem

This work investigates the computational procedures used to obtain global solution to the economic linear optimal control (ELOC) problem. The proposed method employs the generalized Benders decomposition (GBD) algorithm. Compared to the previous branch and bound approach, a naive application of GBD to the ELOC problem will improve computational performance, due to less frequent calls to computationally slow semi-definite programming (SDP) routines. However, the reverse-convex constraints of the original problem will reappear in the relaxed master problem. In response, a convexification of the relaxed master constraints has been developed and proven to preserve global solution characteristics. The result is a multi-fold improvement in computational performance. A technological benefit of decomposing the problem into steady-state and dynamic parts is the ability to utilize nonlinear steady-state models, since the relaxed master problem is free of SDP type constraints and can be solved using any global nonlinear programming algorithm.

Position Location Scheme Using Nonlinear Programming Based on RSSI and DV-Hop

Localization is used in location-aware applications such as navigation, autonomous robotic movement, and asset tracking to position a moving object on a coordinate system. In this paper, a Nonlinear Programming algorithm is proposed based on RSSI and improved DV-Hop algorithm, called NPRDV-Hop. The algorithm makes four major contributions to the localization problem in the wireless sensor networks (WSNs). Firstly, a hop distance is improved, called Hop Distance. This scheme could assure that the most nodes receive the Hop Distance from beacon node who has the least hops between them. This practical localization scheme is relatively high accuracy and low cost for WSNs. Secondly, Heron's formula is introduced as objective function. Thirdly, Gauss distribution is introduced to select RSSI so that the error of distance is little. Lastly, the general problem is considered by the nonlinear programming to solve for the locations of the sensors. Simulation results show that the proposed method can improve location accuracy and coverage without increasing hardware cost of sensor node. The performance of this algorithm is superior to the original DV-Hop algorithm.

Optimization of sub-ambient separation systems with embedded cubic equation of state thermodynamic models and complementarity constraints

A previously developed equation-based flowsheet optimization framework is extended and applied to design sub-ambient separation systems for oxy-fired coal power systems with carbon capture. Unlike most commercial flowsheet design and optimization tools, the proposed methods use exact derivatives and large-scale nonlinear programming algorithms to solve large flowsheet design problems with many degrees of freedom, including the simultaneous design of air separation units (ASUs) and their accompanying multistream heat exchangers. Emphasis is placed on additional model improvements regarding thermodynamic calculations. In order to maintain differentiability, complementarity constraints are used to model switches, including vanishing and reappearing phases. Nevertheless, these complementarity constraints may construct trivial phase equilibrium solutions, and a procedure based on embedded bubble and dew points calculations is proposed to avoid them. Furthermore, additional complementarity constraints for the cubic equation of state model are proposed to ensure correct phase identification in the supercritical region. Finally, the efficacy of these new models are demonstrated by optimization of the CO2 processing unit and compression train for an oxy-fired power plant.

An augmented Lagrangian affine scaling method for nonlinear programming

In this paper, we propose an augmented Lagrangian affine scaling (ALAS) algorithm for general nonlinear programming, for which a quadratic approximation to the augmented Lagrangian is minimized at each iteration. Different from the classical sequential quadratic programming (SQP), the linearization of nonlinear constraints is put into the penalty term of this quadratic approximation, which results in smooth objective of the subproblem and avoids possible inconsistency among the linearized constraints and trust region constraint. By applying affine scaling techniques to handle the strict bound constraints, an active set type affine scaling trust region subproblem is proposed. Through special strategies of updating Lagrange multipliers and adjusting penalty parameters, this subproblem is able to reduce the objective function value and feasibility errors in an adaptive well-balanced way. Global convergence of the ALAS algorithm is established under mild assumptions. Furthermore, boundedness of the penalty parameter is analyzed under certain conditions. Preliminary numerical experiments of ALAS are performed for solving a set of general constrained nonlinear optimization problems in CUTEr problem library.

Advances in sensitivity-based nonlinear model predictive control and dynamic real-time optimization

Recent results in the development of efficient large-scale nonlinear programming (NLP) algorithms have led to fast, on-line realizations of optimization-based methods for nonlinear model predictive control (NMPC) and dynamic real-time optimization (D-RTO), with predictive nonlinear dynamic (e.g., first principle) models. For NMPC, optimization-based controllers are developed that lead to well-understood stability and robustness properties, even for large, complex plant models. The realization of NMPC requires the application of a fast NLP solver for time-critical, on-line optimization, as well as efficient NLP sensitivity tools that require 2–3 orders of magnitude less computation than the NLP solution. This leads to advanced step NMPC (asNMPC), which essentially eliminates computational delay. We also extend these capabilities to dynamic real-time optimization (D-RTO) with more general stage costs that are economically based. This overview also extends input to state stability (ISS) properties for asNMPC to handle active set changes, and also for D-RTO through convex regularizations. Two large scale distillation case studies, based on nonlinear first principle models, are presented that demonstrate the effectiveness of these approaches.

Direct control of the temperature rise in parallel transmission by means of temperature virtual observation points: Simulations at 10.5 Tesla.

A method using parallel transmission to mitigate B1+ inhomogeneity while explicitly constraining the temperature rise is reported and compared with a more traditional SAR-constrained pulse design. Finite difference time domain simulations are performed on a numerical human head model and for a 16-channel coil at 10.5 Tesla. Based on a set of presimulations, a virtual observation point compression model for the temperature rise is derived. This compact representation is then used in a nonlinear programming algorithm for pulse design under explicit temperature rise constraints. In the example of a time-of-flight sequence, radiofrequency pulse performance in some cases is increased by a factor of two compared with SAR-constrained pulses, while temperature rise is directly and efficiently controlled. Pulse performance can be gained by relaxing the SAR constraints, but at the expense of a loss of direct control on temperature. Given the importance of accurate safety control at ultrahigh field and the lack of direct correspondence between SAR and temperature, this work motivates the need for thorough thermal studies in normal in vivo conditions. The tools presented here will possibly contribute to safer and more efficient MR exams.