A New Augmented Lagrangian Objective Penalty Function for Constrained Optimization Problems

We present in this paper new results on the existence of saddle points of augmented Lagrangian functions for constrained nonconvex optimization. Four classes of augmented Lagrangian functions are considered: the essentially quadratic augmented Lagrangian, the exponential-type augmented Lagrangian, the modified barrier augmented Lagrangian, and the penalized exponential-type augmented Lagrangian. We first show that under second-order sufficiency conditions, all these augmented Lagrangian functions possess local saddle points. We then prove that global saddle points of these augmented Lagrangian functions exist under certain mild additional conditions. The results obtained in this paper provide a theoretical foundation for the use of augmented Lagrangians in constrained global optimization. Our findings also give new insights to the role played by augmented Lagrangians in local duality theory of constrained nonconvex optimization.

In this paper, a new augmented Lagrangian function with 4-piecewise linear NCP function is introduce for the solution of nonlinear prgramming problems with equality constrained and ineqality constaianed. It is proved that a solution of the original constrained problem and corresponding values of Lagrange multipliers can be found by solving a uncunstrained minimization of the augmented Lagrange function. Meamwhile, a new Lagrangian multiplier method corresponding with new augmented Lagrangian function is proposed. and this method is implement able and convergent.

In this paper, an evolutionary optimization method, Evolian, is proposed for the general constrained optimization problem, which incorporates the concept of (1) a multi-phase optimization process and (2) constraint scaling techniques to resolve problem of ill-conditioning. In each phase of Evolian, the typical evolutionary programming (EP) is performed using an augmented Lagrangian objective function with a penalty parameter fixed. If there is no improvement in the best objective function in one phase, another phase of Evolian is performed after scaling the constraints and then updating the Lagrange multipliers and penalty parameter. This procedure is repeated until a satisfactory solution is obtained. Computer simulation results indicate that Evolian gives outperforming or at least reasonable results for multivariable heavily constrained function optimization as compared to other evolutionary computation-based methods.

Differential evolution (DE) is a novel evolutionary approach capable of handling non-differentiable, nonlinear and multi-modal objective functions. Previous studies have shown that DE is an efficient, effective and robust evolutionary algorithm, but usually it takes large computational time for optimizing the computationally expensive objective function, therefore it is necessary to find a trade-off between convergence speed and robustness. For this purpose, in this paper, a new DE based on uniform design is presented for solving nonlinear constrained optimization problems. Constraints are handled by embodying them in an augmented Lagrangian function, where the penalty parameters and multipliers are adapted as the execution of the algorithm proceeds. The efficiency of the proposed methodology is illustrated by solving numerous constrained optimization problems that can be found in the literature

Constrained optimization problems are very important as they are encountered in many science and engineering applications. A hybrid method based on modified augmented Lagran- gian multiplier and biogeography-based optimization (BBO) algorithm is proposed to solve con- strained optimization problems. The basic steps of the proposed method are comprised of an outer iteration, in which the Lagrangian multipliers and various penalty parameters are updated using a first-order update scheme, and an inner iteration, in which a nonlinear optimization of the modified augmented Lagrangian function with simple bound constraints is implemented by BBO algorithm. Numerical results show that the proposed method is reliable and efficient for solving constrained optimization problems.

Stochastic optimization methods have been widely employed to find solutions to structural design optimization problems in the past two decades, especially for truss structures. The primary aim of this study is to introduce a design optimization method combining an augmented Lagrangian function and teaching–learning-based optimization for truss and nontruss structural design optimization. The augmented Lagrangian function serves as a constraint-handling tool in the proposed method and converts a constrained optimization problem into an unconstrained one. On the other hand, teaching–learning-based optimization is employed to resolve the transformed, unconstrained optimization problems. Since the proper values of the Lagrangian multipliers and penalty factors are unknown in advance, the proposed method is implemented in an iterative way to avoid the issue of selecting them, i.e. the Lagrangian multipliers and penalty factors are automatically updated according to the violation level of all constraints. To examine the performance of the proposed method, it is applied on a group of benchmark truss optimization problems and a group of nontruss optimization problems of aircraft wing structures. The computational results obtained by the proposed method are compared to the results produced by both other version of teaching–learning-based optimization and stochastic optimization methods.

Augmented Lagrange Hopfield network initialized by quadratic programming for economic dispatch with piecewise quadratic cost functions and prohibited zones

In this paper relationships between Pareto points and saddle points are studied in convex and nonconvex multiple objective programming. The analysis is based on partitioning the index sets of objectives and constraints and splitting the original problem into subproblems having a special structure. The results are based on scalarizations of multiple objective programs and related linear and augmented Lagrangian functions. In the nonconvex case, a saddle point characterization of Pareto points is possible under assumptions that guarantee existence of Pareto points and stability conditions of single objective problems. Essentially, these conditions are not stronger than those in analogous results for single objective programming.

Duality in nonlinear programs using augmented Lagrangian functions

This paper investigates the system achievable rate for the multiple-input multiple-output orthogonal frequency division multiplexing (MIMO–OFDM) system with an energy harvesting (EH) relay. Firstly we propose two protocols, time switching-based decode-and-forward relaying (TSDFR) and a flexible power splitting-based DF relaying (PSDFR) protocol by considering two practical receiver architectures, to enable the simultaneous information processing and energy harvesting at the relay. In PSDFR protocol, we introduce a temporal parameter to describe the time division pattern between the two phases which makes the protocol more flexible and general. In order to explore the system performance limit, we discuss the system achievable rate theoretically and formulate two optimization problems for the proposed protocols to maximize the system achievable rate. Since the problems are non-convex and difficult to solve, we first analyze them theoretically and get some explicit results, then design an augmented Lagrangian penalty function (ALPF) based algorithm for them. Numerical results are provided to validate the accuracy of our analytical results and the effectiveness of the proposed ALPF algorithm. It is shown that, PSDFR outperforms TSDFR to achieve higher achievable rate in such a MIMO–OFDM relaying system. Besides, we also investigate the impacts of the relay location, the number of antennas and the number of subcarriers on the system performance. Specifically, it is shown that, the relay position greatly affects the system performance of both protocols, and relatively worse achievable rate is achieved when the relay is placed in the middle of the source and the destination. This is different from the MIMO–OFDM DF relaying system without EH. Moreover, the optimal factor which indicates the time division pattern between the two phases in the PSDFR protocol is always above 0.8, which means that, the common division of the total transmission time into two equal phases in previous work applying PS-based receiver is not optimal.

In this paper, the classical exact absolute value penalty function method is used for solving a new class of nonconvex nonsmooth optimization problems. Nonconvex nondifferentiable optimization problems with both inequality and equality constraints are considered here, in which not all functions constituting them have the fundamental property of convex functions and most classes of generalized convex functions—namely, a stationary point of such a function is its global minimum. It is proved for such nonconvex optimization problems that there exists a finite threshold of penalty parameters equal to the largest absolute value of a Lagrange multiplier such that, for every penalty parameter exceeding this lower bound, there is the equivalence between an optimal solution in the original constrained minimization problem with ‐invex functions and a minimizer in its associated penalized optimization problem with the exact l1 penalty function. Further, under nondifferentiable ‐invexity assumptions, a characterization of a saddle point of the Lagrange function, defined for the considered constrained optimization problem in terms of minimizers of its associated exact penalized optimization problem with the exact l1 penalty function, is presented.

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.

Three layer control structure is proposed for optimal control of continuous fermentation processes. The start-up optimization problems are solved as a first step for optimization layer building. A steady state optimization problem is solved by a decomposition method using prediction principle. A discrete minimum time optimal control problem with state delay is formulated and a decomposition method, based on an augmented Lagrange's function is proposed to solve it. The problem is decomposed in time domain by a new coordinating vector. The obtained algorithms are used for minimum time optimal control calculation of Baker's Yeast fermentation process.

Exact augmented lagrangian function for nonlinear programming problems with inequality constraints

We propose and analyze a class of penalty-function-free nonmonotone trust-region methods for nonlinear equality constrained optimization problems. The algorithmic framework yields global convergence without using a merit function and allows nonmonotonicity independently for both, the constraint violation and the value of the Lagrangian function. Similar to the Byrd–Omojokun class of algorithms, each step is composed of a quasi-normal and a tangential step. Both steps are required to satisfy a decrease condition for their respective trust-region subproblems. The proposed mechanism for accepting steps combines nonmonotone decrease conditions on the constraint violation and/or the Lagrangian function, which leads to a flexibility and acceptance behavior comparable to filter-based methods. We establish the global convergence of the method. Furthermore, transition to quadratic local convergence is proved. Numerical tests are presented that confirm the robustness and efficiency of the approach.