现代优化方法现代优化方法 (8).pdf
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1、9.3 Augmented Lagrangian MethodQingna Li(BIT)9.3 Augmented Lagrangian Method1/12MotivationConsider the equality constrained case.minxf(x)s.t.ci(x)=0,i E,(1.1)The quadratic penalty method solvesminxf(x)+k2iEc2i(x).(1.2)Quadratic Penalty returns infeasible solutionxk:ci(xk)=0.will lead to ill-conditio
2、ned problem.Qingna Li(BIT)9.3 Augmented Lagrangian Method2/12Augmented Lagrangian FunctionGiven k,k,define the augmented Lagrange functionLA(x;k;k)=f(x)iEici(x)+k2iEc2i(x).xk arg minxLA(x;k;k)gives0 xLA(xk,k;k)=f(xk)ik kci(xk)ci(xk).Comparing with the optimality condition for(1.1)0=xL(xk,)=f(xk)ici(
3、xk),we can geti ki kci(xk),i E.So we update k+1ibyk+1i=ki kci(xk),i E.(1.3)Qingna Li(BIT)9.3 Augmented Lagrangian Method3/12Algorithm 9.3(Augmented Lagrangian Method-EqualityConstranits)Given 0 0,tolerance 0 0,starting points xs0and 0;fork=0,1,2.Find an approximate minimizer xkof LA(,k;k),starting a
4、txsk,and terminating when xLA(xk,k;k)k;if a convergence test for(1.1)is satisfiedstop with approximate solution xk;end(if)Update Lagrangian multipliers using(1.3)to obtain k+1;Choose k+1 k;Let xsk+1=xk;Select tolerance k+1;end(for)Qingna Li(BIT)9.3 Augmented Lagrangian Method4/12Exampleminx1+x2s.t.x
5、21+x22 2=0The optimal solution is(1,1)T,the Lagrangemultiplier is=0.5.The augmented Lagrangian function isLA(x;,)=x1+x2(x21+x22 2)+2(x21+x22 2)2.At k-th iteration,suppose k=0.4,k=1.See theFigure below for LA(x;0.4,1).Qingna Li(BIT)9.3 Augmented Lagrangian Method5/12Figure:Countours of L(x,)for =0.4
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