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Semidefinite optimization (SDO) differs from linear optimization (LO) in that it deals with optimization problems over the cone of symmetric positive semidefinite matrices Sn+ instead of nonnegative vectors. In many cases the objective function is linear and SDO can be interpreted as an extension of LO. It is a branch of convex optimization and contains — besides LO — linearly constrained QP, quadratically constrained QP and — for example — second-order cone programming as special cases.
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Forst, W., Hoffmann, D. (2010). Semidefinite Optimization. In: Optimization—Theory and Practice. Springer Undergraduate Texts in Mathematics and Technology . Springer, New York, NY. https://doi.org/10.1007/978-0-387-78977-4_7
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