Evaluation of Penalty Functions for Optimal Control

Abstract

Abstract:To handle final st ate equality constraints, two ty pe s of pen al ty functions are evaluated a nd compared in solving t hree op timal cont rol pr oblems. Both t he a bsol ute valu e pen al ty func t ion and t he quadra t ic pen al ty fun ction with shift ing terms yielded t he opt imal control policy in each cas e. T he qu adrat ic penal ty fun cti on with shift ing te r ms gave t he optimum more accurately, and t he a pp roach to t he optimum is less oscillatory t han with t he use of abso lute value pen alty fun cti on . In solving for t he optimal control in t he third examp le, t he use of the absolute value penalt y function required a series of runs to get t he performanc e ind ex to wit hin 0.035% of t he opti mum . A furt her advantage in usin g the q uadratic penalty function is obtaining sens it ivi ty informat ion wit h resp ect to t he final state const raints from t he shifti ng te r ms.

Optimization of Fed-Batch Reactors by the Luus−Jaakola Optimization Procedure

“…Therefore, the inequality expressed by eq 10 becomes equality, and we formulate the augmented performance index to be maximized as where a penalty term is added to the performance index in eq 10. In the penalty function we have included the shifting term s. The advantages of such a penalty function including a shifting parameter to handle equality constraints have been recently demonstrated. ,− The shifting parameter s is initially set to zero and then is updated after every pass of the optimization procedure according to where q denotes the pass number in the multipass optimization procedure. At the optimum, the factor −2θ s gives the Lagrange multiplier that yields the sensitivity information, showing the effect of changing the volume constraint x 4 ( t f ) = 200.…”

Optimal Control by Iterative Dynamic Programming with Deterministic and Random Candidates for Control

“…The use of the quadratic penalty function with a shifting term was introduced by Luus as a way of dealing with difficult equality constraint in steady-state optimization. It was found to be efficient in handling equality state constraints in optimal control problems. − …”

“…It was found to be efficient in handling equality state constraints in optimal control problems. [16][17][18] To have direct comparison with the results in the literature, 12,14 we chose the final time to be 63.0 h, and we considered the case when the time interval was divided into 20 stages (P ) 20). Hartig et al 12 compared three optimization methods: IDP, sequential quadratic programming (SQP), and the direct search optimization method of Luus and Jaakola 19 (LJ optimization) in solving this problem and reported that IDP yielded the highest success rate with P ) 20.…”

Handling inequality constraints in direct search optimization

“…However, it should be noted that the approach recommended here is not restricted to this optimization procedure. Details of the LJ optimization procedure in solving a variety of optimization problems are available in the literature (Luus et al 1995, Luus 2000b,c, 2001. The method can be used successfully for optimization of systems of very high dimension (Luus 1998a).…”