A dynamic system model for solving convex nonlinear optimization problems

In this paper, we present a neural network for solving the nonlinear convex programming problem in real time by means of the projection method. The main idea is to convert the convex programming problem into a variational inequality problem. Then a dynamical system and a convex energy function are constructed for resulting variational inequality problem. It is shown that the proposed neural network is stable in the sense of Lyapunov and can converge to an exact optimal solution of the original problem. Compared with the existing neural networks for solving the nonlinear convex programming problem, the proposed neural network has no Lipschitz condition, no adjustable parameter, and its structure is simple. The validity and transient behavior of the proposed neural network are demonstrated by some simulation results.

This paper presents a recurrent neural network for solving nonconvex nonlinear optimization problems subject to nonlinear inequality constraints. First, the p‐power transformation is exploited for local convexification of the Lagrangian function in nonconvex nonlinear optimization problem. Next, the proposed neural network is constructed based on the Karush–Kuhn–Tucker (KKT) optimality conditions and the projection function. An important property of this neural network is that its equilibrium point corresponds to the optimal solution of the original problem. By utilizing an appropriate Lyapunov function, it is shown that the proposed neural network is stable in the sense of Lyapunov and convergent to the global optimal solution of the original problem. Also, the sensitivity of the convergence is analysed by changing the scaling factors. Compared with other existing neural networks for such problem, the proposed neural network has more advantages such as high accuracy of the obtained solutions, fast convergence, and low complexity. Finally, simulation results are provided to show the benefits of the proposed model, which compare to or outperform existing models.

Analog networks for solving convex nonlinear unconstrained programming problems without using gradient information of the objective function are proposed. The one-dimensional net can be used as a building block in multidimensional networks for optimizing objective functions of several variables.

A neural network for solving convex nonlinear programming problems is proposed in this paper. The distinguishing features of the proposed network are that the primal and dual problems can be solved simultaneously, all necessary and sufficient optimality conditions are incorporated, and no penalty parameter is involved. Based on Lyapunov, LaSalle and set stability theories, we prove strictly an important theoretical result that, for an arbitrary initial point, the trajectory of the proposed network does converge to the set of its equilibrium points, regardless of whether a convex nonlinear programming problem has unique or infinitely many optimal solutions. Numerical simulation results also show that the proposed network is feasible and efficient. In addition, a general method for transforming nonlinear programming problems into unconstrained problems is also proposed.

A capable neural network model for solving the maximum flow problem

Recently some specific classes of non-smooth and non-Lipsch-itz convex optimization problems were considered by Yu. Nesterov and H. Lu. We consider convex programming problems with similar smoothness conditions for the objective function and functional constraints. We introduce a new concept of an inexact model and propose some analogues of switching subgradient schemes for convex programming problems for the relatively Lipschitz-continuous objective function and functional constraints. Some class of online convex optimization problems is considered. The proposed methods are optimal in the class of optimization problems with relatively Lipschitz-continuous objective and functional constraints.

The convex hull heuristic is a heuristic for mixed-integer programming problems with a nonlinear objective function and linear constraints. It is a matheuristic in two ways: it is based on the mathematical programming algorithm called simplicial decomposition, or SD (von Hohenbalken in Math Program 13:49–68, 1977), and at each iteration, one solves a mixed-integer programming problem with a linear objective function and the original constraints, and a continuous problem with a nonlinear objective function and a single linear constraint. Its purpose is to produce quickly feasible and often near optimal or optimal solutions for convex and nonconvex problems. It is usually multi-start. We have tested it on a number of hard quadratic 0–1 optimization problems and present numerical results for generalized quadratic assignment problems, cross-dock door assignment problems, quadratic assignment problems and quadratic knapsack problems. We compare solution quality and solution times with results from the literature, when possible.

A numerical multi-level optimization methodology is proposed for determining dextrous workspaces of 3-degree-of-freedom (3-dof) planar parallel manipulators, in which it is required that at any point within the workspace, the manipulator is able to assume any orientation in a specified range. The method starts by finding a single initial point on the boundary of the dextrous workspace. This first stage requires the successive solution of three separate optimization sub-problems, where the evaluation of the objective function for the second problem and the constraint functions in the third problem are determined by the solution of appropriate optimization problems at a lower level. Once the boundary point is identified, further successive points along the dextrous workspace boundary are traced by the application of the so-called chord method. In the latter procedure, the determination of each successive boundary point is also obtained via a constrained optimization problem, where the constraint functions are again evaluated via the solution of an optimization problem at a lower level. The proposed method is illustrated by its successful application to three different manipulator design geometries, and for various ranges of dexterity.

An optimization procedure is developed for the design of structural components under crashworthiness requirements. The optimization procedure consists of first developing a system of parallel neural networks able to reproduce the structural behavior during the crash phenomena. Training and test sets are generated by finite element analyses. Then, to find the optimal configuration, both sequential quadratic programming and genetic algorithms are implemented to evaluate the objective and constraint functions using the response surfaces generated by the neural networks. The procedure is applied for the optimization of helicopter subfloor components. In this application, the objective function is taken equal to the sum of two terms: the crush force efficiency and the specific absorbed energy. The values of maximum and mean forces are constrained to meet crashworthiness requirements, whereas other constraints are applied to define the feasibility of the structural solution. The optimal configuration, obtained measuring orders of magnitude savings in the CPU time, allows an increase of the crush force efficiency equal to 18% and a decrease of the mass equal to 8%.

In this paper, we present a multi-objective fractional probabilistic programming problem where the parameters in the objective functions and right-hand side constraints are multi-choice in nature. Multi-choice multi-objective objective fractional probabilistic programming problem is formulated. The multi-choice parameters in the right-hand side of constraints are random variables following Cauchy distribution. Multi-choice multi-objective fractional probabilistic programming problem is transformed to deterministic equivalent multi-choice multi-objective fractional programming problem using chance-constrained programming problem. Lagrange interpolating polynomial approach is proposed to deal with multi-choice parameters. The multi-choice multi-objective fractional programming problem is transformed to non-linear mixed integer multi-objective programming problem. An \(\epsilon -\) constrained method is applied to find the compromise solutions of non-linear mixed integer multi-objective programming problem using different values of \(\epsilon \). LINGO software is applied to find the optimal solution of single objective non-linear mixed integer programming problem. Finally, a numerical example is given to illustrate the proposed programming problem.KeywordsMulti-choice programming problemMulti-objective programming problemProbabilistic programming problemFractional programming problem\(\epsilon \)-constraint method

In this paper, a special nonlinear bilevel programming problem (nonlinear BLPP) is transformed into an equivalent single objective nonlinear programming problem. To solve the equivalent problem effectively, we first construct a specific optimization problem with two objectives. By solving the specific problem, we can decrease the leader's objective value, identify the quality of any feasible solution from infeasible solutions and the quality of two feasible solutions for the equivalent single objective optimization problem, force the infeasible solutions moving toward the feasible region, and improve the feasible solutions gradually. We then propose a new constraint-handling scheme and a specific-design crossover operator. The new constraint-handling scheme can make the individuals satisfy all linear constraints exactly and the nonlinear constraints approximately. The crossover operator can generate high quality potential offspring. Based on the constraint-handling scheme and the crossover operator, we propose a new evolutionary algorithm and prove its global convergence. A distinguishing feature of the algorithm is that it can be used to handle nonlinear BLPPs with nondifferentiable leader's objective functions. Finally, simulations on 31 benchmark problems, 12 of which have nondifferentiable leader's objective functions, are made and the results demonstrate the effectiveness of the proposed algorithm.

본 논문은 사각 보강 패널에 사용되는 모자형 보강재의 크기 최적화에 대한 결과를 제시하였다. 보강 패널은 1500 ~ 2500 m/s 의 속도를 가지는 발사체와 충돌한다. 모자형 보강재의 크기를 결정하기 위해서 크기 최적화가 수행되었다. 크기 최적화를 수행하기 위해서는 3 개의 함수들로 구성되어 있다. 이 함수들은 목적함수, 제한함수 그리고 설계 함수로 이루어져 있다. 목적 함수는 보강 패널의 1 차 고유진동수의 최대화가 되도록 하는 것이다. 제함 함수는 보강재의 부피가 사각 패널 부피의 10 % 이내가 되도록 하는 것이며, 설계 변수로는 모자형 보강재의 치수가 된다. 최적화된 모자형 보강재를 이용하한 보강 패널을 사용하여 초고속 충돌에 대한 시뮬레이션을 수행하였으며, 최적화된 보강재에 대해서 속도와 운동 에너지 변화에 대한 결과를 얻었다. 보강 패널의 충돌 저감을 평가하기 위해서 발사체의 운동에너지와 속도를 무차원화 계산을 수행하여 비교 분석 하였다.

We consider a production system consisting of >N processing stages. The actual leadtimes at the stages are stochastic. The objective is to determine the planned leadtimes at each stage so as to minimize the expected total inventory costs, tardiness penalties, and a backlog penalty for not meeting demand due date at the last stage. Recursive relationships are used for automatic generation and efficient computation of the objective function. The efficiency of the proposed algorithms allows us to obtain new insights regarding operating policies, leadtime delivery reliability, and production line design. The problem is formulated as a convex non-linear programming problem. The latter is then solved using classical convex optimization algorithms. For the special case of exponentially distributed leadtimes, the objective function is derived in a closed form.

Recently, robust minimum variance (MV) beamforming which optimizes the worst-case performance has been proposed in S.A. Vorobyov et al. (2003), R.G. Lorenz and S.P. Boyd (2005). The worst-case approach, however, might be overly conservative in practical applications. In this paper, we propose a more flexible approach that formulates the robust adaptive beamforming problem as a probability-constrained optimization problem with homogeneous quadratic cost function. Unlike the general probability-constrained problem which can be nonconvex and NP-hard, our problem can be reformulated as a convex nonlinear programming (NLP) problem, and efficiently solved using interior-point methods. Simulation results show an improved robustness of the proposed beamformer as compared to the existing state-of-the-art robust adaptive beamforming techniques

In this paper, we present a high-performance dynamic optimization scheme to solve a class of fractional programming (FP) problems. The main idea is to convert the FP problem into an equivalent convex second-order cone programming problem. A neural network model based on a dynamic model is then constructed for solving the obtained convex programming problem. By employing a credible Lyapunov function approach, it is shown that the proposed neural network model is stable in the sense of Lyapunov and is globally convergent to an exact optimal solution of the original optimization problem. A block diagram of the model is also given. Several illustrative examples are provided to show the efficiency of the proposed method in this manuscript.