Exact augmented lagrangian function for nonlinear programming problems with inequality constraints

A dynamic convexized method for nonconvex mixed integer nonlinear programming

In this paper, a new augmented Lagrangian function is introduced for solving nonlinear programming problems with inequality constraints. The relevant feature of the proposed approach is that, under suitable assumptions, it enables one to obtain the solution of the constrained problem by a single unconstrained minimization of a continuously differentiable function, so that standard unconstrained minimization techniques can be employed. Numerical examples are reported.

The main motivation of this paper is to weaken the conditions that imply the correspondence between the solution of a constrained problem and the unconstrained minimization of a continuously differentiable function. In particular, a new continuously differentiable exact penalty function is proposed for the solution of nonlinear programming problems. Under mild assumptions, a complete equivalence can be established between the solution of the original constrained problem and the unconstrained minimization of this penalty function on a perturbation of the feasible set. This new penalty function and its exactness properties allow us to define globally and superlinearly convergent algorithms to solve nonlinear programming problems. As an example, a Newton-type algorithm is described which converges locally in one iteration in case of quadratic programming problems.

Duality in nonlinear programs using augmented Lagrangian functions

Constrained optimization for microbial fermentation was studied. For optimization, we used not the maximum principle but a nonlinear programming method because of the need to consider many metabolic reactions. In the case of L-lysine fermentation, the optimization problem in L-lysine production was formulated as a nonlinear programming problem. In general, the state equations based on material balances are represented as differential equations, but such equations which are dependent on time can not be applied to a nonlinear programming problem. Therefore, the state equations were made discrete in a time base, and a new single vector which is not dependent on time was substituted. From these formulae, the objective function and the constraints using nonlinear programming problem were defined as the amount of L-lysine produced, and as a metabolic reaction model and empirical equations, respectively. Computer program was developed to solve this constrained nonlinear programming problem. The applied algorithm of the computer programming was a sequential quadratic programming method (SQP method). When the constrained nonlinear programming problem is solved using the SQP method, the maximum amount of L-lysine produced and the optimal feeding rate of L-threonine could be calculated. From the calculated results, it was clear that introduction of the equality and inequality constraints was easy. L-Lysine at a concentration up to 75.3 g/l could be produced when the fermentation was carried out under optimal conditions.

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.

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

The use of parallel-kinematics machines (PKM) for manufacturing operations is attractive because of the high accuracy they can ensure. These robots might perform a task that requires less degrees of freedom than those offered by the robot. This is the case of a robot facing a functional redundancy, which can be exploited to further increase the accuracy of the task, e.g. upon minimizing the condition number of the Jacobian matrix. A practical case study of a spherical manipulator performing a pointing task are reported, to show how posture-optimization can be used as a redundancy-resolution means for functionally redundant PKMs. The kinematics of the machine and the orientation of the pointing task is used to build, respectively, the objective function and the constraint equations. Sequential Quadratic Programming is conducted to solve the nonlinear constrained optimization problem and to find the end-effector pose corresponding to the robot posture of minimum condition number for every direction of a given pointing path. Lastly, the constrained problem is rewritten as one of unconstrained optimization of one objective function in one design variable.

A vector-valued generalized Lagrangian is constructed for a nonlinear multiobjective programming problem. Using the Lagrangian, a multiobjective dual is considered. Without assuming differentiability, weak and strong duality theorems are established using Pareto efficiency.

The paper describes a new approach to the optimal-power-flow problem based on Newton's method which it operates with an augmented Lagrangian function associated with the original problem. The function aggregates all the equality and inequality constraints. The first-order necessary conditions for optimality are reached by Newton's method, and by updating the dual variables and the penalty terms associated with the inequality constraints. The proposed approach does not have to identify the set of binding constraints and can be utilised for an infeasible starting point. The sparsity of the Hessian matrix of the augmented Lagrangian is completely exploited in the computational implementation. Tests results are presented to show the good performance of this approach.

In this paper we consider inequality constrained nonlinear optimization problems where the first order derivatives of the objective function and the constraints cannot be used. Our starting point is the possibility to transform the original constrained problem into an unconstrained or linearly constrained minimization of a nonsmooth exact penalty function. This approach shows two main difficulties: the first one is the nonsmoothness of this class of exact penalty functions which may cause derivative-free codes to converge to nonstationary points of the problem; the second one is the fact that the equivalence between stationary points of the constrained problem and those of the exact penalty function can only be stated when the penalty parameter is smaller than a threshold value which is not known a priori. In this paper we propose a derivative-free algorithm which overcomes the preceding difficulties and produces a sequence of points that admits a subsequence converging to a Karush–Kuhn–Tucker point of the constrained problem. In particular the proposed algorithm is based on a smoothing of the nondifferentiable exact penalty function and includes an updating rule which, after at most a finite number of updates, is able to determine a “right value” for the penalty parameter. Furthermore we present the results obtained on a real world problem concerning the estimation of parameters in an insulin-glucose model of the human body.

A nonlinear programming approach to kinematic shakedown analysis of frictional materials

Towards a Discrete Newton Method with Memory for Largescale Optimization R.H. Byrd, et al. On Regularity for Generalized Systems and Applications M. Castellani, et al. An Algorithm Using Quadratic Interpolation for Unconstrained Derivative Free Optimization A.R. Conn, P.L. Toint. Massively Parallel Solution of Large Scale Network Flow Problems R. De Leone, et al. Correlation Theorems for Nonsmooth Systems V.F. Dem'yanov. Successive Projection Methods for the Solution of Overdetermined Nonlinear Systems M.A. Diniz-Erhardt, J.M. Martinez. On Exact Augmented Lagrangian Functions in Nonlinear Programming G. Di Pillo, S. Lucidi. Spacetransformation Technique: The State of the Art Y.G. Evtushenko, G.Z. Vitali. Semismoothness and Superlinear Convergence in Nonsmooth Optimization and Nonsmooth Equations H. Jiang, et al. On the Solution of the Monotone and Nonmonotone Linear Complementarity Problem by an Infeasible Interior Point Algorithm J. Judice, et al. Ergodic Results in Subgradient Optimization T. Larsson, et al. Protoderivatives and the Geometry of Solution Mappings in Nonlinear Programming A.B. Levy, R.T. Rockafellar. Index.

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.

The present paper is devoted to the application of the space transformation technique for solving linear and nonlinear programming problems. By using a surjective mapping the initial constrained optimization problem is transformed to a problem in a new space with only equality constraints. For the numerical solution of the latter problem we use the generalized gradient-projection method and Newton’s method. After inverse transformation to the initial space we obtain the family of numerical methods for solving optimization problems with equality and inequality constraints. In the linear programming case after some simplification and after choosing a particular exponential transformation function we obtain from the proposed methods the well-known primal-dual interior point linear programming algorithm and the so-called “variation of Kermarkar’s algorithm”.