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Within a SQP- algorithm it can happen that the constraints of the quadratic sub- problems are infeasible. In order to overcome this infeasibilities, a l1 penalty method can be used according to Nocedal and Wright. The quadratic program has then the form: $$ \min_{p,v,w,t} \space f_k+\nabla f(x)^Tp+\frac{1}2 p^THp+\mu \sum_{i\in \mathcal{E}}{(v_i+w_i)}+\mu \sum_{i\in \mathcal{I}}{t_i} $$ $$ s.t. \space \nabla c_i(x)^Tp+c_i(x)=v_i-w_i, \space i\in \mathcal{E} $$ $$ \space \space \space \nabla c_i(x)^Tp+c_i(x)\geq-t_i, i\in \mathcal{I} $$ $$ r,s,t\geq 0 $$ Maybe a trust region constraint of the form $$ ||p||_{\infty}\leq\Delta_k $$ is added to the problem.

Nocedal and Wright state on page 555 that this problem can be solved by a standard quadratic programming algorithm. How is this done? How can the problem be transformed to the "standard" form $$ \min_{p} \space f_k+\nabla f(x)^Tp+\frac{1}2 p^THp $$ $$ s.t. \space \nabla c_i(x)^Tp+c_i(x)=0, \space i\in \mathcal{E} $$ $$ \space \space \space \nabla c_i(x)^Tp+c_i(x)\geq0, i\in \mathcal{I} $$

Or, alternatively, what do the KKT- conditions look like for this problem?

Regards,

Karsten

Karsten
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  • Addressing your concern "How can the problem be transformed to the "standard" form " I don't see any problem. As $\mu$ is fix in the quadratic program, you have the minimization of a quadratic form subject to affine and linear equality constraints. This is, indeed, a standard quadratic problem. – The Pheromone Kid Apr 24 '15 at 08:36

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