Users:Structural Optimization/Optimization Algorithms/Augmented Lagrange Multiplier

Contents 1 Motivation 2 Input Parameters 2.1 Input Example 3 References

Motivation

The Augmented Lagrange Multiplier (ALM) Method is an algorithm for constrained optimization problems utilizing first order gradients and Lagrange multipliers. Like in other penalty methods (c.f. ^[1], ^[2]) the constraint problem is solved by several unconstrained optimization steps. The unconstrained problem is formulated by an augmented Lagrangian function which tends to the original objective close to the constrained optimum.

In contrast to interior point methods like the Method of Feasible Directions the ALM Method does not require a feasible initial design. Moreover it is not seldom that the algorithm works for many optimization steps in the infeasible domain until it reaches feasible designs.

Basically the minimization of the constrained optimization problem is substituted by unconstrained minimization of an augmented Lagrangian function stated as

L_A (x^k, λ, μ, r_p) = f(x^k) + ∑_i[ λ_i ψ_i + r_p ψ_i²] + ∑_j[ μ_j h_j + r_p (h_i)²]

where the inequality constraints are formulated in the function ψ.

During optimization the penalty parameter is usually increased until a specified maximum is reached.

r_p^k+1 = r_p^k * GAMMA, r_p^k+1 ≤ PENALTY_MAX

The algorithmic implementation of the augmented Lagrange multiplier method is complex and lengthy. It is referred to the references for more information.

Input Parameters

Block headline
Parameter	Values, Default(*)	Description
OPT-CTR	int : CG_NAND	ID and identifier of optimization algorithm.
Common Compulsory Parameters, valid for all optimization algorithms
FILTER	OPT-FILTER int, int, ....	One or more filter function IDs.
OBJ	OPT-RESPONSE_FCT int, int, ...	One or more response functions that are considered as objective.
OUTPUT	PC-OUT int	The Output object.
DOMAIN	EL-DOMAIN int	The respective domain on which the optimization problem is defined.
REGULARIZATION	EL-REGULARIZATION int	The regularization object.
LINE_SEARCH	OPT-LINE_SEARCH int	The line search object.
CONVERGENCE_CONTROL	OPT-CONVERGENCE int	The convergence checker.
Common Optional Parameters, valid for all optimization algorithms
DESIGN_SPACE_BOUNDS	ND-SET int, int, ...	One or more node set IDs that define the boundary of the design space.
RESTART_DATA_FREQ	int	Frequency of restart output,
Specific parameters for ALM method
CON	OPT-RESPONSE_FCT int, int, ...	One or more response functions used as constraints.
PENALTY	float	Initial penalty parameter rp.
PENALTY_MAX	float	Maximum penalty parameter rp.
GAMMA	float	Multiplier to update penalty parameter rp.

Input Example

Example of a complete input block:

OPT-CTR 1 : ALM_NAND

  ! compulsory parameter
  FILTER=OPT-FILTER 1
  OBJ=OPT-RESPONSE_FCT 1
  CON=OPT-RESPONSE_FCT 2, 3, 4
  OUTPUT=PC-OUT 1
  DOMAIN = EL-DOMAIN 1
  REGULARIZATION = EL-REGULARIZATION 1
  LINE_SEARCH = OPT-LINE_SEARCH 1
  CONVERGENCE_CONTROL = OPT-CONVERGENCE 1

  ! optional parameter
  DESIGN_SPACE_BOUNDS = ND-SET 6

  !********** ALM-PARAMETERS ******
  PENALTY = 1.0
  PENALTY_MAX = 5.0
  GAMMA = 1.25

References

1. ↑ R. Haftka, Z. Gürdal, Elements of Structural Optimization, Kluwer Academic Publishers, 1984
2. ↑ G. Vanderplaats, Numerical Optimization Techniques for Engineering Design: With Applications, McGraw Hill, 1992

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