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Chapter 8

Nonlinear Programming with Constraints

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Methods for Solving NLP Problems

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*)(**)( xhxf (a) Where 414.1/1* is called the Lagrange multiplier for the constraint h = 0

)21,21(* x ; see Fig. E 8.1a

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Note that there are n + m equations in the n + m unknowns x and λ

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1 2Minimize : ( )f x xx

2 21 2Subject to : ( ) 25g x x x

By the Lagrange multiplier method.

Solution: The Lagrange function is

2 21 2 1 2( , ) ( 25)L u x x x x x u

The necessary conditions for a stationary point are

2 11

2 0L

x uxx

1 22

2 0L

x uxx

2 21 2 25

Lx x

u

2 21 2(25 ) 0u x x

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Penalty functions for handling equality constraints

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for handling inequality constraints

Note g must be >0 ; r 0

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The logarithmic barrier function formulation for m constraints is

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Use xc = 2 yc = 2 for linearization

(step bounds)

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Quadratic Programming (QP)

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8.3 QUADRATIC PROGRAMMING

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Use of Quadratic Programming to Design Multivariable Controllers

(Model Predictive Control)

• Targets (set points) selected by real-time optimization software based on current operating and economic conditions

• Minimize square of deviations between predicted future outputs and specific reference trajectory to new targets using QP

• Framework handles multiple input, multiple output (MIMO) control problems with constraints on manipulated and controlled variables. Dynamics obtained from transfer function model.

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Successive Quadratic Programming

• Considered by some to be the best general nonlinear programming algorithm

• Repetitively approximates nonlinear objective function with quadratic function and nonlinear constraints with linear constraints

• Uses line search rather than QP step for each iteration• Inequality constrained Quadratic Programming (IQP)

keeps all inequality constraints• Equality constrained Quadratic Programming (EQP) only

keeps equality constraints by utilizing and active set strategy

• SQP is an Infeasible Path method

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solve for ,x

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Generalized Reduced Gradient (GRG)

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• sequential simplex• conjugate gradient• Newton’s method• Quasi-Newton

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we have to supplement blast furnace gas with fuel oil, but we want to minimize the purchase of fuel oil.

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define: X1 = amount of fuel oil used in generator 1X2 = amount of fuel oil used in generator 2X3 = amount of BFG used in generator 1X4 = amount of BFG used in generator 2P1 = mw output of generator 1P2 = mw output of generator 2

range of operation of generator 1 and generator 218 ≤ P1 ≤ 3014 ≤ P2 ≤ 25

Fuel effects in the generators are additive (can operateon either BFG or fuel oil)

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10 units of BFG are available (on the average):

1 unit BFG = Btu equivalent of 1 ton/hr. fuel oil.

We need 50 mw power at all times.

5021 PP

Experimental data needed?

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Mathematical Statement

1 2

fuel oil to gen 1 fuel oil to gen 2

Min f x x

a. operating ranges 18 ≤ P1 ≤ 30& requirements 14 ≤ P2 ≤ 25

P1 + P2 = 50

b. availability of x3 + x4

blast furnace gas

c. operating P11 (x1) fuel oilcharacteristics P12 (x3) BFG

P21 (x2) fuel oil P22 (x4)

BFG

gen 1

gen 2

2221212111 PPPPPP

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BFG addmay

wesince sufficientnot isequation this

00145.15186.4609.1 21111 XXP

22212

422

221

12111

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BFG )(

oil fuel )(

1generator in reqt.)(

PPP

XP

XP

PPP

XBFGP

fcn of burners, heat transfer characteristics (convex functions)

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Solution

NLP 4 ineq. const.piece-wise LP 6 eq. const.

1

2

3.05

30

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optf

P

P

No fuel oil is used in generator 1.In generator 2, fuel oil provides 58%of the power (rest is BFG).

heat transfer characteristics may change, or BFG mayvary w.r.t. time (on-line solution)

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