Feasible direction

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Given a feasible point LaTeX: x, a direction vector, say LaTeX: d, is feasible if there exists LaTeX: s > 0 such that LaTeX: x + sd is feasible. The method of feasible directions is designed to choose a feasible direction at each iteration, along which (as LaTeX: s becomes positive) the objective value improves. Such directions exist for continuous mathematical programs (where the line segment LaTeX: [x, x + sd] is feasible for some LaTeX: s > 0) unless LaTeX: x is a local optimum. (Note: with nonlinear constraints, there is typically no feasible direction according to this (classical) definition. See the generalized reduced gradient method.)

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