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It's not the gradient but the new point that should satisfy the constraint |
The direction-finding subproblem and the update rule did not comply with each other. Either x_k +s in D in the subproblem and x_k+1 <-- x_k + \alpha s in the update or s in D in the subproblem and x_k+1 <-- x_k + \alpha (s - x_k) in the update are used. See Jaggi (2013) |
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:'''Step 1.''' ''Direction-finding subproblem:'' Find <math>\mathbf{s}_k</math> solving
::Minimize <math> \mathbf{s}^T \nabla f(\mathbf{x}_k)</math>
::Subject to <math>
:''(Interpretation: Minimize the linear approximation of the problem given by the first-order [[Taylor series|Taylor approximation]] of <math>f</math> around <math>\mathbf{x}_k \!</math> constrained to stay within <math>\mathcal{D}</math>.)''
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