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Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks to optimize (minimize or maximize) a multivariate quadratic function subject to linear constraints on the variables.
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Given a real matrix M and vector q, the linear complementarity problem LCP(q, M) seeks vectors z and w which satisfy the following constraints: w , z ⩾ 0 , {\displaystyle w,z\geqslant 0,} (that is, each component of these two vectors is non-negative)
Since the quadratic form is a scalar quantity, = (). Next, by the cyclic property of the trace operator, [ ()] = [ ()]. Since the trace operator is a linear combination of the components of the matrix, it therefore follows from the linearity of the expectation operator that
Let q be a quadratic form defined on an n-dimensional real vector space. Let A be the matrix of the quadratic form q in a given basis. This means that A is a symmetric n × n matrix such that =, where x is the column vector of coordinates of v in the chosen basis.
Convex quadratically constrained quadratic programs can also be formulated as SOCPs by reformulating the objective function as a constraint. [4] Semidefinite programming subsumes SOCPs as the SOCP constraints can be written as linear matrix inequalities (LMI) and can be reformulated as an instance of semidefinite program. [4]
A comparison of the convergence of gradient descent with optimal step size (in green) and conjugate vector (in red) for minimizing a quadratic function associated with a given linear system. Conjugate gradient, assuming exact arithmetic, converges in at most n steps, where n is the size of the matrix of the system (here n = 2).
Non-negative least squares problems turn up as subproblems in matrix decomposition, e.g. in algorithms for PARAFAC [2] and non-negative matrix/tensor factorization. [3] [4] The latter can be considered a generalization of NNLS. [1]