Stochastic optimisation with inequality constraints using simultaneous perturbations and penalty functions

In this paper, we present a stochastic approximation algorithm based on penalty function method and a simultaneous perturbation gradient estimate for solving stochastic optimization problems with general inequality constraints. We also present a very general convergence result for the proposed algorithm.

We introduce a twice differentiable augmented Lagrangian for nonlinear optimization with general inequality constraints and show that a strict local minimizer of the original problem is an approximate strict local solution of the augmented Lagrangian. A novel augmented Lagrangian method of multipliers (ALM) is then presented. Our method is originated from a generalization of the Hestenes-Powell augmented Lagrangian, and is a combination of the augmented Lagrangian and the interior-point technique. It shares a similar algorithmic framework with existing ALMs for optimization with inequality constraints, but it can use the second derivatives and does not depend on projections on the set of inequality constraints. In each iteration, our method solves a twice continuously differentiable unconstrained optimization subproblem on primal variables. The dual iterates, penalty and smoothing parameters are updated adaptively. The global and local convergence are analyzed. Without assuming any constraint qualification, it is proved that the proposed method has strong global convergence. The method may converge to either a Karush-Kuhn-Tucker (KKT) point or a singular stationary point when the converging point is a minimizer. It may also converge to an infeasible stationary point of nonlinear program when the problem is infeasible. Furthermore, our method is capable of rapidly detecting the possible infeasibility of the solved problem. Under suitable conditions, it is locally linearly convergent to the KKT point, which is consistent with ALMs for optimization with equality constraints. The preliminary numerical experiments on some small benchmark test problems demonstrate our theoretical results.

On an exact penalty function method for nonlinear mixed discrete programming problems and its applications in search engine advertising problems

For smooth optimization problem with equality constraints, new continuously differentiable penalty function is derived. It is proved exact in the sense that local optimizers of a nonlinear program are precisely the optimizers of the associated penalty function under some nondegeneracy assumption. It is simple in the sense that the penalty function only includes the objective function and constrained functions, and it doesn’t include their gradients. This is achieved by augmenting the dimension of the program by a variable that controls the weight of the penalty terms.

Targeting the adjustment of the errors-in-variables (EIV) model with equality and inequality constraints, a general solution that is similar to the classical least square adjustment is proposed based on the penalty function and the weight in measurement. Firstly, we take the equality constraints as inequality constraints that do not satisfy the constraint conditions and construct the penalty functions of equality and inequality constraints, respectively. Thus, the inequality constrained optimization problem is transformed into an unconstrained optimization problem. Then the detailed calculation formula and approximate accuracy evaluation formula of the general solution are deduced. The iteration formula of the general solution is easy regarding comprehension and applicable in implementation. It can not only solve the EIV model with equality and inequality constraints respectively, but also address the EIV model with equality and inequality constraints simultaneously. In addition, it can promote the Gauss–Markov (G-M) model with equality and inequality constraints. Finally, three examples (i.e., equality constraints, inequality constraints and those with equality and inequality constraints) are validated, indicating that the general solution is effective and feasible. The results show that the general solution is effective and feasible.

The penalty methods constitute a family of particularly interesting algorithms of the two points of view: the simpleness of principle and the practical efficiency. They are considered as ill-conditioned because the penalty parameter rk tends to infinity, as k tends infinity. It is what led to us to introduce what we call augmented Lagrangian to regularize the solutions. Namely, to avoid the numerical instability of the classical penalty method. Another favors of the augmented Lagrangian is that by presence of the term λi gi , the exact solution of the problem can be determined without making aim rk towards the infinity, contrary to the penalty method, where it has the effect of diverting the packaging of the problem to be resolved. The using of augmented Lagrangian is considered as an improvement of the penalty methods. It avoids having to use too big parameters of penalties. Besides, the fact of adding the quadratic term rk (g+)2 in the Lagrangian will improve the properties of convergence of the algorithms of duality in this paper. In this paper, we study some augmented Lagrangian algorithms applied to convex nondifferentiable optimization problems and prove their convergence.

In this paper we propose two methods for smoothing a nonsmooth square-root exact penalty function for inequality constrained optimization. Error estimations are obtained among the optimal objective function values of the smoothed penalty problem, of the nonsmooth penalty problem and of the original optimization problem. We develop an algorithm for solving the optimization problem based on the smoothed penalty function and prove the convergence of the algorithm. The efficiency of the smoothed penalty function is illustrated with some numerical examples, which show that the algorithm seems efficient.

In this study, we introduce a new smoothing approximation to the non-differentiable exact penalty functions for inequality constrained optimization problems. Error estimations are investigated between non-smooth penalty function and smoothed penalty function. In order to demonstrate the effectiveness of proposed smoothing approach the numerical examples are given.

This chapter introduces two very important concepts in constrained nonlinear optimization. These are penalty and augmented Lagrangian concepts. The idea is that both these concepts replace the original problem by a sequence of sub-problems in which the constraints are expressed by terms added to the objective function. The penalty concept is implemented in two different methods. The quadratic penalty method adds to the objective function a multiple of the square of the violation of each constraint and solves a sequence of unconstrained optimization sub-problems. Simple and enough intuitive, this approach has some important deficiencies. The nonsmooth exact penalty method, on the other hand, solves a single unconstrained optimization problem. In this approach, a popular function is the l1 penalty function. The problem with this method is that the nonsmoothness may create complications in numerical implementations. Finally, the second concept is the multiplier method or the augmented Lagrangian method, which explicitly uses Lagrange multiplier estimates in order to avoid the ill-conditioning of the quadratic penalty method.

The non-monotonic reasoning system based on penalty function was originally introduced by Pinkas in 1995. The main idea of this method is to get unknown knowledge via the voting method according to some known knowledge. However, the penalty function method, when applied to the problem of intelligent route planning, cannot effectively reflect human intelligence. A new voting principle and a new method to construct its penalty function are presented. Experimental results show that, in some real applications, our new penalty function is more reasonable and effective than that of Pinkas.

Chapter 5 - Nonquadratic Penalty Functions — Convex Programming

An economic load dispatch model developed for both wind turbines and thermal generators is presented in this paper. In the model, the wind power is considered as a constraint due to random nature of wind speed. This optimization problem is solved to obtain optimal power outputs of thermal generators, while taking into account of given constraints. All these optimal outputs and available wind power must satisfy the load generation balance of the power system. The closed form solution of this model is not possible due to the inequality constraints and this problem is solved by introducing a penalty function or barrier function in the model to take account of inequality constraints. The feasible ranges of optimum solutions are obtained by using the developed models based on penalty function and interior point methods.

When we have prior information of approximate and subjective nature other than those contained in the time series, that is knowledge not directly referable to exact linear constraints, we propose to use the penalty function method to tackle the problem of estimation of a linear statistical model alternatively to the mixed estimation proposed by Theil and Goldberger. This method bases itself on the minimization of a mathematical function known as penalty function where constraint expressed in a quadratic form is linearly combined with an objective function by some coefficients known as penalty parameters. The particular functional form utilized for the constraint can be seen as a way to catch all the uncertainty included in this type of prior knowledge. The algorithm that derives from the application of this method depends on the penalty parameters that can be interpreted in two different ways: a) as weights assigned to the constraint in the minimization procedure. In this case the result can be seen as a generalization of the ridge estimator. b) as variances of a stochastic constraint. By this way the algorithm is similar to the mixed estimator but, unlike this one, it has not unknown parameters in its formulation and so it can be seen as a feasible estimator. Whatever the interpretation of this parameters is, the penalty function method produces an algorithm that is a flexible instrument both from a theoretical viewpoint and a practical one.

The purpose of this paper is twofold. First we consider a class of nondifferentiable penalty functions for constrained Lipschitz programs and then we show how these penalty functions can be employed to solve a constrained Lipschitz program. The penalty functions considered incorporate a barrier term which makes their value go to infinity on the boundary of a perturbation of the feasible set. Exploiting this fact it is possible to prove, under mild compactness and regularity assumptions, a complete correspondence between the unconstrained minimization of the penalty functions and the solution of the constrained program, thus showing that the penalty functions are exact according to the definition introduced in [17]. Motivated by these results, we propose some algorithm models and study their convergence properties. We show that, even when the assumptions used to establish the exactness of the penalty functions are not satisfied, every limit point of the sequence produced by a basic algorithm model is an extended stationary point according to the definition given in [8]. Then, based on this analysis and on the one previously carried out on the penalty functions, we study the consequence on the convergence properties of increasingly demanding assumptions. In particular we show that under the same assumptions used to establish the exactness properties of the penalty functions, it is possible to guarantee that a limit point at least exists, and that any such limit point is a KKT point for the constrained problem.

We propose a reduced-space interior-point approach to nonlinear optimization problems with general inequality constraints. It is an extension of the quasi-sequential approach to dynamic optimization of large-scale systems. Inequality constraints are formed by adding slack variables to an equality constrained barrier (interior-point) problem which is solved by a range space step and a null space step in every iteration. Mathematical derivations and computation schemes are presented. We take a highly nonlinear parameter estimation problem as an example to demonstrate the effectiveness of this approach. The result is compared with the full space approach in terms of overall CPU time and number of iterations.