In this paper we use measure theory to solve a wide range of the nonlinear programming problems. First, we transform a nonlinear programming problem to a classical optimal control problem with no restriction on states...
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In this paper we use measure theory to solve a wide range of the nonlinear programming problems. First, we transform a nonlinear programming problem to a classical optimal control problem with no restriction on states and controls. The new problem is modified into one consisting of the minimization of a special linear functional over a set of Radon measures;then we obtain an optimal measure corresponding to functional problem which is then approximated by a finite combination of atomic measures and the problem converted approximately to a finite-dimensional linear programming. Then by the solution of the linear programming problem we obtain the approximate optimal control and then, by the solution of the latter problem we obtain an approximate solution for the original problem. Furthermore, we obtain the path from the initial point to the admissible solution.
We show that the effects of finite-precision arithmetic in forming and solving the linear system that arises at each iteration of primal-dual interior-point algorithms for nonlinear programming are benign, provided th...
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We show that the effects of finite-precision arithmetic in forming and solving the linear system that arises at each iteration of primal-dual interior-point algorithms for nonlinear programming are benign, provided that the iterates satisfy centrality and feasibility conditions of the type usually associated with path-following methods. When we replace the standard assumption that the active constraint gradients are independent by the weaker Mangasarian-Fromovitz constraint qualification, rapid convergence usually is attainable, even when cancellation and roundoff errors occur during the calculations. In deriving our main results, we prov a key technical result about the size of the exact primal-dual step. This result can be used to modify existing analysis of primal-dual interior-point methods for convex programming, making it possible to extend the superlinear local convergence results to the nonconvex case.
An algorithm for minimizing a nonlinear function subject to nonlinear inequality constraints is described. It applies sequential quadratic programming techniques to a sequence of barrier problems, and uses trust regio...
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An algorithm for minimizing a nonlinear function subject to nonlinear inequality constraints is described. It applies sequential quadratic programming techniques to a sequence of barrier problems, and uses trust regions to ensure the robustness of the iteration and to allow the direct use of second order derivatives. This framework permits primal and primal-dual steps, but the paper focuses on the primal version of the new algorithm. An analysis of the convergence properties of this method is presented.
Homotopy methods are globally convergent under weak conditions and robust;however, the efficiency of a homotopy method is closely related with the construction of the homotopy map and the path tracing algorithm. Diffe...
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Homotopy methods are globally convergent under weak conditions and robust;however, the efficiency of a homotopy method is closely related with the construction of the homotopy map and the path tracing algorithm. Different homotopies may behave very different in performance even though they are all theoretically convergent. In this paper, a spline smoothing homotopy method for nonconvex nonlinear programming is developed using cubic spline to smooth the max function of the constraints of nonlinear programming. Some properties of spline smoothing function are discussed and the global convergence of spline smoothing homotopy under the weak normal cone condition is proven. The spline smoothing technique uses a smooth constraint instead of m constraints and acts also as an active set technique. So the spline smoothing homotopy method is more efficient than previous homotopy methods like combined homotopy interior point method, aggregate constraint homotopy method and other probability one homotopy methods. Numerical tests with the comparisons to some other methods show that the new method is very efficient for nonlinear programming with large number of complicated constraints.
nonlinearly constrained nonlinear programming (NLC-NLP) problems arise in various real-world decision-making fields, such as financial engineering, urban planning, supply chain management, and power system control. Th...
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nonlinearly constrained nonlinear programming (NLC-NLP) problems arise in various real-world decision-making fields, such as financial engineering, urban planning, supply chain management, and power system control. They are usually large-scale because of having to consider massive variables and constraints. Solving NLC-NLP problems by employing common algorithms (e.g., gradient projection method (GPM)) is usually computationally-expensive, which challenges common organizations in solving large-scale NLC-NLP problems. To address this issue, an option is to adopt cloud computing for help. However, this raises security concerns since real-world NLC-NLP problems may carry sensitive information. Although previous secure outsourcing algorithms try to protect sensitive information, they still let cloud service tenants bear heavy computation burden. In this paper, we develop a practical secure outsourcing algorithm for using the GPM to solve large-scale NLC-NLP problems. To be more prominent, to accelerate computations and avoid possible memory overflowing, we parallelize the developed algorithm. We implement the developed algorithm on the Amazon Elastic Compute Cloud (EC2) and a laptop, and also offer extensive experiment results to show that the developed algorithm can reduce the tenant's computing time significantly.
作者：
MCKINNEY, DCLIN, MDMckinney DC
UNIV TEXAS DEPT CIVIL ENGN ENVIRONM & WATER RESOURCES ENGN PROGRAM AUSTIN TX 78712 USA
An optimal aquifer remediation design model employing a nonlinear programming algorithm was developed to find the minimum cost design of a pump-and-treat aquifer remediation system. The mixed-integer nonlinear program...
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An optimal aquifer remediation design model employing a nonlinear programming algorithm was developed to find the minimum cost design of a pump-and-treat aquifer remediation system. The mixed-integer nonlinear programming model includes the discontinuous fixed costs of system construction and installation as well as operation and maintenance. The fu;ed cost terms in the objective function have been approximated by continuous functions of the decision variables using a polynomial penalty coefficient method resulting in a nonlinear programming formulation of an otherwise mixed-integer nonlinear programming model. Results of applying the new polynomial penalty coefficient method to an example design problem show that a combined well field and treatment process model that includes fixed costs has a significant impact on the design and cost of aquifer remediation systems, reducing system costs by using fewer, larger flow rate wells. Previous pump-and-treat design formulations have resulted in systems with numerous, low flow rate wells due to the use of simplified cost functions that do not exhibit economies of scale or fixed costs. The polynomial penalty coefficient method results were compared to two alternative approximate mixed-integer nonlinear programming methods for solving optimal aquifer remediation design problems, the pseudo-integer method and the exponential penalty coefficient method. The polynomial penalty coefficient method obtains the same solutions and performs as well as or better than the exponential penalty coefficient method. The polynomial penalty coefficient method almost always results in better, less expensive designs and requires significantly less computer time than the pseudo-integer method.
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 horiz...
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An efficient new SQP algorithm capable of solving large-scale problems is described. It generates descent directions for an l(1) plus log-barrier merit function and uses a line-search to obtain a sufficient decrease o...
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An efficient new SQP algorithm capable of solving large-scale problems is described. It generates descent directions for an l(1) plus log-barrier merit function and uses a line-search to obtain a sufficient decrease of this function. The unmodified exact Hessian matrix of the Lagrangian function is normally used in the QP subproblem, but this is set to zero if it fails to yield a descent direction for the merit function. The QP problem is solved by an interior-point method using an inexact Newton approach, iterating to an accuracy just sufficient to produce a descent direction in the early stages and tightening the accuracy as we approach a solution. We prove finite termination of the algorithm, at an epsilon -optimal Fritz-John point if feasibility is attained. We also show that if any iterate is close enough to an isolated connected subset of local minimizers, then the iterates converge to this subset. The rate of convergence is Q-quadratic if the subset is an isolated minimizer which satis es a second-order sufficiency condition, but Q-quadratic convergence to an epsilon -optimal point can still be achieved without any conditions beyond Lipschitz continuity of second-order derivatives. The implementation SQPIPM is designed for problems with many degrees of freedom and is shown to perform well compared with other codes on a range of standard problems.
In this paper, we establish a nonlinear Lagrangian algorithm for nonlinear programming problems with inequality constraints. Under some assumptions, it is proved that the sequence of points, generated by solving an un...
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In this paper, we establish a nonlinear Lagrangian algorithm for nonlinear programming problems with inequality constraints. Under some assumptions, it is proved that the sequence of points, generated by solving an unconstrained programming, convergents locally to a Kuhn-Tucker point of the primal nonlinear programming problem.
Recently developed Newton and quasi-Newton methods for nonlinear programming possess only local convergence properties. Adopting the concept of the damped Newton method in unconstrained optimization, we propose a step...
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Recently developed Newton and quasi-Newton methods for nonlinear programming possess only local convergence properties. Adopting the concept of the damped Newton method in unconstrained optimization, we propose a stepsize procedure to maintain the monotone decrease of an exact penalty function. In so doing, the convergence of the method is globalized.
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