Users:Structural Optimization/Optimization Algorithms/Augmented Lagrange Multiplier
Contents |
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
LA (xk, λ, μ, rp) = f(xk) + ∑i[ λi ψi + rp ψi2] + ∑j[ μj hj + rp (hi)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.
rpk+1 = rpk * GAMMA, rpk+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 FILTFUNC=OPT-LINKFUNC 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 FILTERING_CONCEPT = TOP DESIGN_SPACE_BOUNDS = ND-SET 6 !********** ALM-PARAMETERS ****** PENALTY = 5.0 PENALTY_MAX = 20.0 GAMMA = 1.25
References
- ↑ R. Haftka, Z. Gürdal, Elements of Structural Optimization, Kluwer Academic Publishers, 1984
- ↑ G. Vanderplaats, Numerical Optimization Techniques for Engineering Design: With Applications, McGraw Hill, 1992
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