Lagrange multipliers and nonlinear programming

The modified barrier function approach for large-scale optimization

Inspired by the Lagrangian multiplier method with quadratic penalty function, which is widely used in Nonlinear Programming Theory, a Lagrange-type nonlinear programming neural network whose equilibria coincide with KKT pairs of the underlying nonlinear programming problem was devised with minor modification in regard to handling inequality constraints[1,2]. Of course, the structure of neural network must be elaborately conceived so that it is asymptotically stable. Normally this aim is not easy to be achieved even for the simple nonlinear programming problems. However, if the penalty parameters in these neural networks are taken as control variables and a control law is found to stabilize it, we may reasonably conjecture that the categories of solvable nonlinear programming problems will be greatly increased. In this paper, the conditions stabilizing the Lagrange-type neural network are presented and control-Lyapunov function approach is used to synthesize the adjusting laws of penalty parameters.KeywordsPenalty ParameterNonlinear Programming ProblemGlobal Asymptotic StabilityFeedback StabilizationHierarchical OptimizationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The regularization of a nonlinear program is exact if all solutions of the regularized problem are also solutions of the original problem for all values of the regularization parameter below some positive threshold. In Deng (Pac J Optim 8(1):27–32, 2012), we show that, for a given nonlinear program, the regularization is exact if and only if the Lagrangian function of a certain selection problem has a saddle point, and the regularization parameter threshold is inversely related to the Lagrange multiplier associated with the saddle point. The results in Deng (Pac J Optim 8(1):27–32, 2012) not only provide a fresh perspective on exact regularization but also extend the main results in Friedlander and Tseng (SIAM J Optim 18:1326–1350, 2007) on a characterization of exact regularization of a convex program to that of a nonlinear (not necessarily convex) program. In this paper, we will examine inner-connections among exact regularization, normal cone identity, and the existence of a weak sharp minimum for certain associated nonlinear programs. Along the way, we illustrate by examples, how to obtain both new results and reproduce many existing results from a fresh perspective.

One of the important chemicals is propylene glycol. Propylene glycol is used in various sectors of the national economy, including the production of saturated polyester resins, in the preparation of pharmaceutical and cosmetic preparations, in the manufacture of paints, in the food industry, etc. In this regard, the development of algorithms to ensure the technological process of production of propylene glycol in optimal regimes is an urgent problem for the development of the chemical and petrochemical industry. The aim of this work is to calculate the optimal working regime of the hydrator for technological process of propylene glycol production based on mathematical models, including the Lagrangian function. The process of obtaining propylene glycol is carried out in accordance with the given temperature and pressure in the hydrator. As for the quantitative and qualitative indicators of the production of propylene glycol, the necessary conditions are set. For example, the density of commercial propylene glycol produced should be 0.981–1.036 g/cm³. Quantity and quality of the propylene glycol obtained in the process depends on the predetermined amount of propylene oxide, temperature and pressure in the hydrator. When solving the problem of optimizing the static regime of the technological process, such parameters of propylene glycol quality as quantity and density were taken into account. These indicators were taken into account in accordance with coordinates of states and constraint conditions as nonlinear functions. Taking into account the above, problem of optimal control of the parameters in the hydrator was formulated taking into account the real limitations imposed on the input and control parameters in the technological scheme (propylene glycol consumption, propylene glycol density, pressure in the hydrator, temperature in the hydrator, amount of steam and propylene oxide supplied in the mixer). To solve this problem, an algorithm was developed based on a model including the Lagrangian function. The optimum operating regime for the hydrator of the process for the production of propylene glycol was calculated on the basis of nonlinear programming. To solve the optimization problem, the coordinates of the saddle point of the Lagrangian function were determined. The optimum values of temperature, pressure and water-oxypropylene ratio in the hydrator, consumption and density of propylene glycol are found. For the obtained optimal data, it is recommended to use the nonlinear programming method, taking into account the obtained coordinates of the saddle point of the Lagrangian function.

Recent Developments in Nonlinear Programming

A deterministic global optimization algorithm based on augmented Lagrangian is proposed for solving the nonlinear programming with cone constraints, which is often encountered in technical design and operational research. At each outer iteration, the method requires the -global minimization of the augmented Lagrangian of problems with cone constraints, where. The convergence to an-global minimizer of the original problem is proved. Key words: Global optimization, augmented Lagrangians, cone optimization, algorithms.

Directional derivatives can be very effectively used in the solution of classical optimization problems (Braswell and Marban 1971, 1972). They measure the rate of change of a function in a given direction and give an approximation of a function in the neighbourhood of a stationary point. From this approximation general necessary and sufficient conditions for any classical optimization problem can be established. Feasibility can be assured by choosing the components of the direction vectors in such a way that the constraints will be satisfied. The approach is used to derive the sufficient conditions of the equality constrained optimization problem, when Lagrange multipliers are used in the solution of the problem. The conditions are equivalent to those previously developed but are computationally and pedagogically advantageous.

This chapter contains sections titled: Effect of Constraints on the Solution Space of Nonlinear Programs Methods for Solving Nonlinear Programs Direct Substitution Method Method of Constrained Variations Lagrange Multipliers: Kuhn-Tucker Conditions Interpretation of Lagrange Multipliers Constraint Qualification Testing Kuhn-Tucker Conditions Nature of Solution of Nonlinear Programs Quadratic Programs Feasible Directions Gradient Projection Methods��-��Generalized Reduced Gradient Method Sequential Linear Programming Method Sequential Quadratic Programming (SQP) Synthesis of Lagrange Multipliers from Solution Rate of Convergence Summary Exercise Problems ]]>

We develop a model to determine an optimal investment strategy to improve the performance of undergraduate students in the US. Our model has three parts: In the first part, we collect data about the focus of other foundations’ investment by subjects and locations. We consider the charitable identity of the Goodgrant as well. Then we set out to decide our focus, which is to invest more on those schools with more minority races, lower educational performance, higher debt ratio and so on. In this part, we also classify the data into two groups, one for school selecting, and another for ROI determining. In the second part, as a data extraction, we build an efficient and intuitive model to rank the candidate schools in accordance with the correlation of our focus, using the PCA method. After that, the top 50 schools are selected as our target schools. In the third part, we make a key assumption that the social utility of a school has logarithmic relationship with the earnings of graduated students and the graduation rate. More over, we create a parameter k to denote the marginal rate of substitution (MRS) between the two factors above. After that, we come to define the ROI function of each target school as the incremental utility. We further discuss how to devise the best strategy with several methods. At last, we choose the improved PSO algorithm based on augmented Lagrange function. This algorithm is a typical method to solve the multivariable optimization problem with constraint conditions. Then we offer a recommending list by the cumulative ROI in five years. What’s more, our model is broad enough to accommodate any non-linear constraint optimization problem. Finally, we change the numerical value of parameter k to examine the sensitivity of our investment strategy. The result shows that our model is robust.

The circuit tuning problem is best approached by means of gradient-based nonlinear optimization algorithms. For large circuits, gradient computation can be the bottleneck in the optimization procedure. Traditionally, when the number of measurements is large relative to the number of tunable parameters, the direct method is used to repeatedly solve the associated sensitivity circuit to obtain all the necessary gradients. Likewise, when the parameters outnumber the measurements, the adjoint method is employed to solve the adjoint circuit repeatedly for each measurement to compute the sensitivities. In this paper, we propose the adjoint Lagrangian method, which computes all the gradients necessary for augmented-Lagrangian-based optimization in a single adjoint analysis. After the nominal simulation of the circuit has been carried out, the gradients of the merit function are expressed as the gradients of a weighted sum of circuit measurements. The weights are dependent on the nominal solution and on optimizer quantities such as Lagrange multipliers. By suitably choosing the excitations of the adjoint circuit, the gradients of the merit function are computed via a single adjoint analysis, irrespective of the number of measurements and the number of parameters of the optimization. This procedure requires close integration between the nonlinear optimization software and the circuit simulation program. The adjoint Lagrangian formulation has been implemented in the JiffyTune tool which optimizes delay, area, slew (transition time) and power measurements by adjusting transistor widths and wire sizes. Speedups of over 35x have been realized in the gradient computation procedure by using the adjoint Lagrangian formulation, leading to speedups of up to 2.5x in the overall optimization procedure. Perhaps more importantly, these speedups have rendered feasible the tuning of large circuits. A circuit with 6,900 transistors was optimized in under two hours of CPU time.

A minimum cost production model which can be used to account economical energy exchange between utilities and evaluate production cost saving in multiple area interconnected systems is proposed in this paper. A fast probabilistic production simulation model is used to determine the thermal incremental cost of each system separately. Based on the thermal incremental cost of each subsystem, a nonlinear optimization model which minimizes the total production cost of the whole interconnected systems is used to determine the economical energy exchanges in the interconnected systems. Then total production cost and cost saving are for whole interconnected systems and each system. The minimum cost principle based on Lagrange multipliers is proposed to solve this nonlinear optimization problem. The tie line capacity limit and energy loss are included in the energy exchange process. Hourly loads are used in the model. The thermal incremental cost of each system and the energy exchanges between interconnected systems are evaluated hour by hour. Then a period summary report is produced. The proposed minimum cost model has been incorporated into FEPSS (Fast Electricity Production Simulation System) package. The Pacific Power Corporation, Australia has used this model to study power system interconnections in Southeast Australia. >

The addition of equality and inequality constraints in generalized predictive control (GPC) strategies switches the classical problem of quadratic cost function minimization into a nonlinear optimization. Some approaches have already been developed, which consider the Lagrange multipliers and gradient optimization. This paper presents the reformulation of GPC under constraints into an adequate form for linear matrix inequalities (LMI) solvers. The optimization of the reformulated problem is performed by linear convex programming. Moreover, it is shown that this new presentation of constrained GPC provides an original flexibility for adding further constraints to the problem. Finally, some simulation results are given and compared to previous approaches of this structure.

Mathematical formulation and modeling of combustion processes is an important tool in the understanding of this phenomenon. Determination of equilibrium temperature and composition is often the first stage in calculation of combustion characteristics. There are number of different techniques for simulation of combustion process. In this study a basic model has been developed based on the minimization of Gibb's free energy to simulate the combustion processes. A nonlinear mathematical optimization has been developed based on Lagrange multipliers and solved using Quasi‐Newton method written in MathCAD environment. The effect of various parameters such as initial temperature, pressure, and equivalence ratio on the equilibrium temperature and composition has been investigated. In contrast with the equilibrium constant method, obtained results out of this work show that the maximum flame temperature for normal paraffins occurs around the equivalence ratio of 1.05. In addition, from environmental point of view, effects of different conditions on the emission of pollutants in a combustion process have been studied. The simulation results showed that the NO and NO2 emission rates pass through a maximum point and there is an optimum point where the NOCO emission could be minimized. Copyright © 2010 Curtin University of Technology and John Wiley & Sons, Ltd.

ABSTRACTAn objective function for a dual model of nonlinear programming problems is an implicit function with respect to Lagrangian multipliers. This study aims to address separable convex programming problems. An explicit expression with respect to Lagrangian multipliers is derived for the dual objective function. The exact solution of the dual model can be achieved because an explicit objective function is more exact than an approximated objective function. Then, a set of improved Lagrangian multipliers can be used to obtain the optimal solution of the original nonlinear programming model. A corresponding dual programming and explicit model (DP-EM) method is proposed and applied to the structural topology optimization of continuum structures. The solution efficiency of the DPEM is compared with the dual sequential quadratic programming (DSQP) method and method of moving asymptotes (MMA). The results show that the DP-EM method is more efficient than the DSQP and MMA.

An excellent airfoil with a high lift-to-drag ratio may decrease oil consumption and enhance the voyage. Based on NACA 0012, an improved airfoil is explored in this paper. The class/shape function transformation has been proved to be a good method for airfoil parameterization, and in this paper it is modified to improve imitation accuracy. The computational fluid dynamics method is applied to obtain numerically the aerodynamic parameters of the parameterized airfoil, and the result is proved credible by comparison with available experimental data in the open literature. A polynomial-based response surface model and the uniform Latin hypercube sampling method are employed to decrease computational cost. Finally, the nonlinear programming by quadratic Lagrangian method is utilized to modify the multi-island genetic algorithm, which has an improved optimization effect than the method used on its own. The obtained result shows that the modified class/shape function transformation method produces a better imitation of an airfoil in the nose and tail regions than the original method, and that it will satisfy the tolerance zone of the model in a wind tunnel. The response surface model based on the uniform Latin hypercube sampling method gives an accurate prediction of the lift-to-drag ratio with changes in the design variables. The numerical result of the flow around the airfoil shows reasonable agreement with the experimental data graphically and quantitatively. Ultimately, an airfoil with better capacity than the original one is acquired using the multi-island genetic algorithm based nonlinear programming by quadratic Lagrangian optimization method. The pressure contours and lift-to-drag ratio along with the attack angle have been compared with those of the original airfoil, and the results demonstrate the strength of the optimized airfoil. The process for exploring an improved airfoil through parameterization to optimization is worth referencing in future work.