Method For Nonlinear Programming Problems Research Articles

The Application of the Piecewise Linear Method for Non-Linear Programming Problems in Ride-Hailing Assignment Based on Service Level, Driver Workload, and Fuel Consumption

Ride-hailing services have grown rapidly, presenting challenges such as increased traffic congestion, inefficient driver workload distribution, and environmental concerns like higher fuel consumption and emissions. This study develops a non-linear ride-hailing assignment model addressing these issues by considering service level, driver workload, and fuel consumption. A piecewise linear method was employed to handle a non-linear programming model, and the method was modified to function autonomously without operator intervention. The model’s performance was evaluated using a publicly accessible dataset of taxi trips in Manhattan, focusing on indicators such as passenger waiting time, driver workload distribution, and fuel consumption. Numerical simulations demonstrated significant improvements: a 15% reduction in average passenger waiting time, a 20% improvement in balancing driver workloads, and a 10% decrease in overall fuel consumption, contributing to reduced emissions and environmental impact. The modified piecewise linear method proved effective in optimizing ride-hailing assignments, providing a more efficient and sustainable solution. The model also showed robustness in handling large datasets, ensuring scalability and applicability to various urban settings. These findings highlight the model’s potential to enhance operational efficiency and promote sustainability in ride-hailing services. By integrating considerations for service level, driver workload, and fuel consumption, the model offers a holistic approach to addressing the key challenges faced by the ride-hailing industry. This study provides valuable insights for future ride-hailing development and implementations of ride-hailing systems, promoting practices that are both efficient and environmentally friendly.

A spline smoothing homotopy method for nonlinear programming problems with both inequality and equality constraints

A spline smoothing homotopy method for nonlinear programming problems with both inequality and equality constraints

Global convergence analysis of the aggregate constraint homotopy method for nonlinear programming problems with both inequality and equality constraints

In this paper, we construct appropriate aggregate mappings and a new aggregate constraint homotopy (ACH) equation by converting equality constraints to inequality constraints and introducing two variable parameters. Then, we propose an ACH method for nonlinear programming problems with inequality and equality constraints. Under suitable conditions, we obtain the global convergence of this ACH method, which makes us prove the existence of a bounded smooth path that connects a given point to a Karush–Kuhn–Tucker point of nonlinear programming problems. The numerical tracking of this path can lead to an implementable globally convergent algorithm. A numerical procedure is given to implement the proposed ACH method, and the computational results are reported.

A new exact penalty function method for nonlinear programming problems

A new exact penalty function method for nonlinear programming problems

On the Convergence Properties of a Second-Order Augmented Lagrangian Method for Nonlinear Programming Problems with Inequality Constraints

The objective of this paper is to conduct a theoretical study on the convergence properties of a second-order augmented Lagrangian method for solving nonlinear programming problems with both equality and inequality constraints. Specifically, we utilize a specially designed generalized Newton method to furnish the second-order iteration of the multipliers and show that when the linear independent constraint qualification and the strong second-order sufficient condition hold, the method employed in this paper is locally convergent and possesses a superlinear rate of convergence, although the penalty parameter is fixed and/or the strict complementarity fails.

The Flattened Aggregate Constraint Homotopy Method for Nonlinear Programming Problems with Many Nonlinear Constraints

The aggregate constraint homotopy method uses a single smoothing constraint instead ofm-constraints to reduce the dimension of its homotopy map, and hence it is expected to be more efficient than the combined homotopy interior point method when the number of constraints is very large. However, the gradient and Hessian of the aggregate constraint function are complicated combinations of gradients and Hessians of all constraint functions, and hence they are expensive to calculate when the number of constraint functions is very large. In order to improve the performance of the aggregate constraint homotopy method for solving nonlinear programming problems, with few variables and many nonlinear constraints, a flattened aggregate constraint homotopy method, that can save much computation of gradients and Hessians of constraint functions, is presented. Under some similar conditions for other homotopy methods, existence and convergence of a smooth homotopy path are proven. A numerical procedure is given to implement the proposed homotopy method, preliminary computational results show its performance, and it is also competitive with the state-of-the-art solver KNITRO for solving large-scale nonlinear optimization.

Inexact restoration and adaptive mesh refinement for optimal control

A new adaptive mesh refinement algorithm is proposed for solving Eulerdiscretization of state- and control-constrained optimal controlproblems. Our approach is designed to reduce the computational effortby applying the inexact restoration (IR) method, a numerical methodfor nonlinear programming problems, in an innovative way. The initialiterations of our algorithm start with a coarse mesh, which typicallyinvolves far fewer discretization points than the fine mesh over whichwe aim to obtain a solution. The coarse mesh is then refinedadaptively, by using the sufficient conditions of convergence of theIR method. The resulting adaptive mesh refinement algorithm isconvergent to a fine mesh solution, by virtue of convergence of the IRmethod. We illustrate the algorithm on a computationally challengingconstrained optimal control problem involving a container crane.Numerical experiments demonstrate that significant computationalsavings can be achieved by the new adaptive mesh refinement algorithmover the fixed-mesh algorithm. Conceivably owing to the small numberof variables at start, the adaptive mesh refinement algorithm appearsto be more robust as well, i.e., it can find solutions with a muchwider range of initial guesses, compared to the fixed-mesh algorithm.

Parallel Iterative Methods for Nonlinear Programming Problems

In this paper, we present two parallel multiplicative algorithms for convex programming. If the objective function is differentiable and convex on the positive orthant of , and it has compact level sets and has a locally Lipschitz continuous gradient, we prove these algorithms converge to a solution of minimization problem. For the proofs there are essentially used the results of sequential methods shown by Eggermont[1].

A revised particle swarm optimization based discrete Lagrange multipliers method for nonlinear programming problems

In this paper, a new algorithm for solving constrained nonlinear programming problems is presented. The basis of our proposed algorithm is none other than the necessary and sufficient conditions that one deals within a discrete constrained local optimum in the context of the discrete Lagrange multipliers theory. We adopt a revised particle swarm optimization algorithm and extend it toward solving nonlinear programming problems with continuous decision variables. To measure the merits of our algorithm, we provide numerical experiments for several renowned benchmark problems and compare the outcome against the best results reported in the literature. The empirical assessments demonstrate that our algorithm is efficient and robust.

Revision of a Floating-Point Genetic Algorithm GENOCOP V for Nonlinear Programming Problems

In this paper, focusing on general nonlinear programming problems, we attempt to propose a general and high- performance approximate solution method for them. In recent years, S. Koziel et al. have proposed a floating-point ge- netic algorithm, GENOCOP V, as a general approximate solution method for nonlinear programming problems and showed its efficiency, there are left some shortcomings of the method. In this paper, incorporating ideas to cope with these shortcomings, we propose a revised GENOCOP V (RGENOCOP V). Furthermore, we show the efficiency of the pro- posed method RGENOCOP V by comparing it with two existing methods, RGENOCOP III and GENOCOP V through the application of them into the numerical examples.

Solving the Karush–Kuhn–Tucker system of a nonconvex programming problem on an unbounded set

In the papers [G.C. Feng, B. Yu, Combined homotopy interior point method for nonlinear programming problems, in: H. Fujita, M. Yamaguti (Eds.), Advances in Numerical Mathematics; Proceedings of the Second Japan–China Seminar on Numerical Mathematics, in: Lecture Notes in Numerical and Applied Analysis, vol. 14, Kinokuniya, Tokyo, 1995, pp. 9–16; G.C. Feng, Z.H. Lin, B. Yu, Existence of an interior pathway to a Karush–Kuhn–Tucker point of a nonconvex programming problem, Nonlinear Analysis 32 (1998) 761–768; Z.H. Lin, B. Yu, G.C. Feng, A combined homotopy interior point method for convex programming problem, Applied Mathematics and Computation 84 (1997) 193–211], a combined homotopy interior method was presented and global convergence results obtained for nonconvex nonlinear programming when the feasible set is bounded and satisfies the so called normal cone condition. However, for when the feasible set is not bounded, no result has so far been obtained. In this paper, a combined homotopy interior method for nonconvex programming problems on the unbounded feasible set is considered. Under suitable additional assumptions, boundedness of the homotopy path, and hence global convergence, is proven.

On the complexity of a combined homotopy interior method for convex programming

In [G.C. Feng, Z.H. Lin, B. Yu, Existence of an interior pathway to a Karush–Kuhn–Tucker point of a nonconvex programming problem, Nonlinear Anal. 32 (1998) 761–768; G.C. Feng, B. Yu, Combined homotopy interior point method for nonlinear programming problems, in: H. Fujita, M. Yamaguti (Eds.), Advances in Numerical Mathematics, Proceedings of the Second Japan–China Seminar on Numerical Mathematics, Lecture Notes in Numerical and Applied Analysis, vol. 14, Kinokuniya, Tokyo, 1995, pp. 9–16; Z.H. Lin, B. Yu, G.C. Feng, A combined homotopy interior point method for convex programming problem, Appl. Math. Comput. 84 (1997) 193–211.], a combined homotopy was constructed for solving non-convex programming and convex programming with weaker conditions, without assuming the logarithmic barrier function to be strictly convex and the solution set to be bounded. It was proven that a smooth interior path from an interior point of the feasible set to a K–K–T point of the problem exists. This shows that combined homotopy interior point methods can solve the problem that commonly used interior point methods cannot solve. However, so far, there is no result on its complexity, even for linear programming. The main difficulty is that the objective function is not monotonically decreasing on the combined homotopy path. In this paper, by taking a piecewise technique, under commonly used conditions, polynomiality of a combined homotopy interior point method is given for convex nonlinear programming.

Optimal search for a moving target with no time information maximizing the expected reward

This paper investigates a search problem for a moving target on a network in which any time information of the target position is not available to a searcher. The searcher has to distribute the limited amount of search efforts on a search space to detect the target, knowing only route information of target paths but not time information about when the target passes there. On detection of the target, the searcher gains some value but expends search cost. There have been few papers which mathematically deal with such a search model without any time information of the target position so far. We formulate the search-efforts- optimizing problem under the expected reward criterion as a convex programming problem and obtain necessary and sufficient conditions for optimal solutions. Using the conditions, a new algorithm is proposed to give an optimal solution. It is shown that the algorithm has the high efficiency for computational time and the robustness for the size of problems comparing with some well-known methods for non-linear programming problems: the gradient projection method and the multiplier method, by numerical examinations. We also elucidate some properties of the optimal solution by the sensitivity analysis of system parameters.

A trust region method for semi-infinite programming problems

We present a new successive quadratic programming (SQP) approach for semi-infinite programming problems with a trust region technique. Numerical methods for solving semi-infinite programming problems can be divided into continuous methods and discretization methods. We begin with a trust region method for nonlinear programming problems which possesses a fast and global convergence property and obviates the Maratos effect which is an unfavourable phenomenon that sometimes occurs for general SQP-type approaches. Then we apply the method to discretized semi-infinite programming problems by utilizing an L exact penalty function and epsilon-most-active constraints. The L exact penalty function is, in fact, essential for continuous methods for semiinfinite programming problems so as to maintain continuity of the exact penalty function, and enables the use of epsilon-most-active constraints in discretized semi-infinite programming problems. The results of preliminary computational experiments demonstrate the effectiveness of our approach for discretized semi-infinite programming problems.

A quadratically convergent scaling newton’s method for nonlinear programming problems

Similar to [6], we present a Q-quadratically convergent scaling Newton's method for nonlinear programming problems. At each iteration, the multiplier variables and slack variable generated by this algorithm all keep in positive orthant of Rm, we only compute a Newton step and determine the step length with at most 2 m division operations, where m is the number of constraints. In Section 4. three examples also support the Q-quadratic convergence

A simple modification of dantzig-wolfe decomposition

We describe a simple modification of the Dintzig-wolfe decomposition method for nonlinear programming problems, consisting of including an additional column in the master problem, which immediately improves the bound obtained. We investigate the consequences of this modification and can directly give the optimal primal solution and one optimal dual solution of the modified master problem. Furthermore we show that the dual solution is always degenerated and give an explicit expression of all the optimal dual solutions. Finally we propose a new hybrid method combining Dantzig-Wolfe decomposition with an improvement step of subgradient type, that generates an additional cut for the master problem in each interation

On a Lagrangian penalty function method for nonlinear programming problems

We modify a Lagrangian penalty function method proposed in [4] for constrained convex mathematical programming problems in order to obtain a geometric rate of convergence. For nonconvex problems we show that a special case of the algorithm in the above paper is still convergent without coercivity and convexity assumptions.

Mixed coordination method for non-linear programming problems with separable structures

Static optimization with linear equality constraints and separable structures is studied by using the mixed coordination method. The idea is to relax equality constraints via Lagrange multipliers, and create a hierarchy where the Lagrange multipliers and part of the decision variables are selected as high-level variables. The method was proposed about ten years ago with a simple high-level updating scheme. We show that the solution of the high-level problem is a saddle point, and the simple updating scheme has a linear convergence rate under appropriate conditions. To obtain faster convergence, the modified Newton method is adopted at the high level. There are two difficulties associated with this approach. One is how to obtain the hessian matrix in determining the Newton direction, since second-order derivatives of the objective function with respect to all high-level variables are needed. The second is when to stop in performing a line search along the Newton direction, as the high-level problem is a maxmini problem looking for a saddle point. In this paper, the hessian matrix is obtained by using a kind of sensitivity analysis. The line search stopping criterion, on the other hand, is based on the norm of the gradient vector. Extensive numerical testings show that our approach performs much better than the simple high-level updating scheme. Since the low level consists of a set of independent subproblems, this method is well suited for parallel implementation in solving large-scale problems. Simulated parallel-processing results show that our method outperforms the one-level Lagrange relaxation method for all the test problems. Furthermore, since convexification terms can be added while maintaining the separability of low-level subproblems, the method is very promising for non-convex problems.

Sensitivity Analysis for Equilibrium Network Flow

Direct application of existing sensitivity analysis methods for nonlinear programming problems or for variational inequalities to nonlinear programming or variational inequality formulations of the equilibrium traffic assignment problem is not feasible, since, in general, the solution to the equilibrium traffic assignment problem does not satisfy the uniqueness conditions required by the sensitivity analysis methods. This paper presents an approach for sensitivity analysis of equilibrium traffic assignment problems in which an equivalent restricted problem is developed which has the desired uniqueness properties; the existing methods are applied to this restricted problem to calculate the derivatives of the equilibrium arc flows with respect to perturbations of the cost functions and of the trip table. These derivatives are then shown to be equivalent to the derivatives of the original unrestricted equilibrium traffic assignment problem; therefore, the method yields the desired sensitivity analysis results.

A Family of Descent Functions for Constrained Optimization

In order to achieve a robust implementation of methods for nonlinear programming problems, it is necessary to devise a procedure which can be used to test whether or not a prospective step would yield a “better” approximation to the solution than the current iterate. In this paper, we present a family of descent or merit functions which are shown to be compatible with local Q-superlinear convergence of Newton and quasi-Newton methods. A simple algorithm is used to verify that good descent and convergence properties are possible using this merit function.