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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). In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-semidefinite.
A sum-of-squares optimization program is an optimization problem with a linear cost function and a particular type of constraint on the decision variables. These constraints are of the form that when the decision variables are used as coefficients in certain polynomials, those polynomials should have the polynomial SOS property.
Download QR code; Print/export ... In mathematics, positive semidefinite may refer to: Positive semidefinite function ... Cookie statement; Mobile view;
There are two main relaxations of QCQP: using semidefinite programming (SDP), and using the reformulation-linearization technique (RLT). For some classes of QCQP problems (precisely, QCQPs with zero diagonal elements in the data matrices), second-order cone programming (SOCP) and linear programming (LP) relaxations providing the same objective value as the SDP relaxation are available.
In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector, where is the row vector transpose of . [1] More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector , where denotes the ...
In contrast with the complex case, a positive-semidefinite operator on a real Hilbert space may not be symmetric. As a counterexample, define A : R 2 → R 2 {\displaystyle A:\mathbb {R} ^{2}\to \mathbb {R} ^{2}} to be an operator of rotation by an acute angle φ ∈ ( − π / 2 , π / 2 ) . {\displaystyle \varphi \in (-\pi /2,\pi /2).}
This implies that at a local minimum the Hessian is positive-semidefinite, and at a local maximum the Hessian is negative-semidefinite. For positive-semidefinite and negative-semidefinite Hessians the test is inconclusive (a critical point where the Hessian is semidefinite but not definite may be a local extremum or a saddle point).
Since is positive semidefinite, all elements in are non-negative. Since the product of two unitary matrices is unitary, taking = one can write = = which is the singular value decomposition. Hence, the existence of the polar decomposition is equivalent to the existence of the singular value decomposition.