L1 Exact Penalty Function Research Articles

On Smoothing l1 Exact Penalty Function for Constrained Optimization Problems

This article introduces a smoothing technique to the l1 exact penalty function. An application of the technique yields a twice continuously differentiable penalty function and a smoothed penalty problem. Under some mild conditions, the optimal solution to the smoothed penalty problem becomes an approximate optimal solution to the original constrained optimization problem. Based on the smoothed penalty problem, we propose an algorithm to solve the constrained optimization problem. Every limit point of the sequence generated by the algorithm is an optimal solution. Several numerical examples are presented to illustrate the performance of the proposed algorithm.

Efficient solution of quadratically constrained quadratic subproblems within the mesh adaptive direct search algorithm

The mesh adaptive direct search algorithm (MADS) is an iterative method for constrained blackbox optimization problems. One of the optional MADS features is a versatile search step in which quadratic models are built leading to a series of quadratically constrained quadratic subproblems. This work explores different algorithms that exploit the structure of the quadratic models: the first one applies an l1-exact penalty function, the second uses an augmented Lagrangian and the third one combines the former two, resulting in a new algorithm. It is notable that this latter approach is uniquely suitable for quadratically constrained quadratic problems. These methods are implemented within the NOMAD software package and their impact are assessed through computational experiments on 65 analytical test problems and 4 simulation-based engineering applications.

An improved smoothing technique-based control vector parameterization method for optimal control problems with inequality path constraints

Summary An improved control vector parameterization (CVP) method is proposed to solve optimal control problems with inequality path constraints by introducing the l1 exact penalty function and a novel smoothing technique. Both the state and control variables are allowed to appear explicitly in the inequality path constraints simultaneously. By applying the penalty function and smoothing technique, all the inequality path constraints are firstly reformulated as non-differentiable penalty terms and incorporated into the objective function. Then, the penalty terms are smoothed by using a novel smooth function, leading to a smooth optimal control problem with no inequality path constraints. With discretizing the control space, a corresponding nonlinear programming (NLP) problem is derived, and error between the NLP problem and the original problem is discussed. Results reveal that if the smoothing parameter is sufficiently small, the solution of the NLP problem is approximately equal to the original problem, which shows the convergence of the proposed method. After clarifying some theories of the proposed approach, a concomitant numerical algorithm is put forward with furnishing the updating rules of both the penalty parameter and smoothing parameter. Simulation examples verify the advantages of the proposed method for tackling nonlinear optimal control problems with different inequality path constraints. Copyright © 2016 John Wiley & Sons, Ltd.

S<i>l</i><sub>1</sub>QP Based Algorithm with Trust Region Technique for Solving Nonlinear Second-Order Cone Programming Problems

In this paper, we propose an algorithm based on Fletcher’s Sl1QP method and the trust region technique for solving Nonlinear Second-Order Cone Programming (NSOCP) problems. The Sl1QP method was originally developed for nonlinear optimization problems with inequality constraints. It converts a constrained optimization problem into an unconstrained problem by using the l1 exact penalty function, and then finds an optimum by solving approximate quadratic programming subproblems successively. In order to apply the Sl1QP method to the NSOCP problem, we introduce an exact penalty function with respect to second-order cone constraints and reformulate the NSOCP problem as an unconstrained optimization problem. However, since each subproblem generated by the Sl1QP method is not differentiable, we reformulate it as a second-order cone programming problem whose objective function is quadratic and constraint functions are affine. We analyze the convergence property of the proposed algorithm, and show that the generated sequence converge to a stationary point of the NSOCP problem under mild assumptions. We also confirm the efficiency of the algorithm by means of numerical experiments.

Smoothing approximation to [formula omitted] exact penalty function for inequality constrained optimization

In this paper, we propose a method to smooth l1 exact penalty function for inequality constrained optimization. It is shown that an approximate global solution of the original problem can be obtained by searching a global solution of the smoothed penalty problem. Under some mild conditions, the method based on our smoothing function is shown to be globally convergent. Some numerical examples are given to illustrate the applicability of the present smoothing method.

THE l1 PENALTY FUNCTION METHOD FOR NONCONVEX DIFFERENTIABLE OPTIMIZATION PROBLEMS WITH INEQUALITY CONSTRAINTS

In this paper, some new results on the l1 exact penalty function method are presented. A simple optimality characterization is given for the nonconvex differentiable optimization problems with inequality constraints via the l1 exact penalty function method. The equivalence between sets of optimal solutions in the original mathematical programming problem and its associated exact penalized optimization problem is established under suitable r-invexity assumption. The penalty parameter is given, above which this equivalence holds. Furthermore, the equivalence between a saddle point in the considered nonconvex mathematical programming problem with inequality constraints and a minimizer in its penalized optimization problem with the l1 exact penalty function is also established.

Computational Experience with a Safeguarded Barrier Algorithm for Sparse Nonlinear Programming

We describe an enhanced version of the primal-dual interior point algorithm in Lasdon, Plummer, and Yu (ORSA Journal on Computing, vol. 7, no. 3, pp. 321–332, 1995), designed to improve convergence with minimal loss of efficiency, and designed to solve large sparse nonlinear problems which may not be convex. New features include (a) a backtracking linesearch using an L1 exact penalty function, (b) ensuring that search directions are downhill for this function by increasing Lagrangian Hessian diagonal elements when necessary, (c) a quasi-Newton option, where the Lagrangian Hessian is replaced by a positive definite approximation (d) inexact solution of each barrier subproblem, in order to approach the central trajectory as the barrier parameter approaches zero, and (e) solution of the symmetric indefinite linear Newton equations using a multifrontal sparse Gaussian elimination procedure, as implemented in the MA47 subroutine from the Harwell Library (Rutherford Appleton Laboratory Report RAL-95-001, Oxfordshire, UK, Jan. 1995). Second derivatives of all problem functions are required when the true Hessian option is used. A Fortran implementation is briefly described. Computational results are presented for 34 smaller models coded in Fortran, where first and second derivatives are approximated by differencing, and for 89 larger GAMS models, where analytic first derivatives are available and finite differencing is used for second partials. The GAMS results are, to our knowledge, the first to show the performance of this promising class of algorithms on large sparse NLP's. For both small and large problems, both true Hessian and quasi- Newton options are quite reliable and converge rapidly. Using the true Hessian, INTOPT is as reliable as MINOS on the GAMS models, although not as reliable as CONOPT. Computation times are considerably longer than for the other 2 solvers. However, interior point methods should be considerably faster than they are here when analytic second derivatives are available, and algorithmic improvements and problem preprocessing should further narrow the gap.

A PROJECTED INDEFINITE DOGLEG PATH METHOD FOR EQUALITY CONSTRAINED OPTIMIZATION

In this paper, we propose a 2-step trust region indefinite dogleg path method for the solution of nonlinear equality constrained optimization problems. The method is a globally convergent modification of the locally convergent Fontecilla method and an indefinite dogleg path method is proposed to get approximate solutions of quadratic programming subproblems even if the Hessian in the model is indefinite. The dogleg paths lie in the null space of the Jacobian matrix of the constraints. An l1 exact penalty function is used in the method to determine if a trial point is accepted. The global convergence and the local two-step superlinear convergence rate are proved. Some numerical results are presented.

Discontinuous piecewise linear optimization

A theoretical framework and a practical algorithm are presented to solve discontinuous piecewise linear optimization problems dealing with functions for which theridges are known. A penalty approach allows one to consider such problems subject to a wide range of constraints involving piecewise linear functions. Although the theory is expounded in detail in the special case of discontinuous piecewiselinear functions, it is straightforwardly extendable, using standard nonlinear programming techniques, tononlinear (discontinuous piecewise differentiable) functions. The descent algorithm which is elaborated uses active-set and projected gradient approaches. It is a generalization of the ideas used by Conn to deal with nonsmoothness in thel1 exact penalty function, and it is based on the notion ofdecomposition of a function into a smooth and a nonsmooth part. The constrained case is reduced to the unconstrained minimization of a (piecewise linear)l1 exact penalty function. We also discuss how the algorithm is modified when it encounters degenerate points. Preliminary numerical results are presented: the algorithm is applied to discontinuous optimization problems from models in industrial engineering. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

An exact penalty function for semi-infinite programming

This paper introduces a global approach to the semi-infinite programming problem that is based upon a generalisation of the l1 exact penalty function. The advantages are that the ensuing penalty function is exact and the penalties include all violations. The merit function requires integrals for the penalties, which provides a consistent model for the algorithm. The discretization is a result of the approximate quadrature rather than an a priori aspect of the model.