Sentence examples for for the equivalent problem from inspiring English sources

For the equivalent problem, the Lagrangian function is given by ( lleft({u}_d^k,lambda right)={u_d^k}^{dagger }Q{u}_d^k+lambda left 1-{u_d^k}^{dagger }F{u}_d^kright) ).

Moreover, he discussed descent methods for solving the equivalent problem (see Lemma 5 in Section 2).

Nevertheless, we note that the equivalent problem (see [26], p. 67) for the definition of equivalent problems), obtained considering the change of variables P S, DT ( k ) = ( 2 b S, DT ( k ) − 1 ) / η S, D ( k ), where b S, DT ( k ) = lo g 2 1 + P S, DT ( k ) η S, D ( k ), is a convex optimization problem.

Since Gk,R has been determined previously for inter-pair interference cancellation and is known, the optimal solution {Gk,L, F k } for the uplink equivalent problem in (20) and (21) (see [15]) can be used to construct the optimal decoder and precoder {G k  = Gk,LGk,R, F k } of the k th pair in the uplink framework.

Since Fk,L has been determined previously for inter-pair interference cancellation and is known, the optimal solution {G k, Fk,R} for the downlink equivalent problem in (16) and (17) (see [15]) can be used to construct the optimal decoder and precoder G k, F k F k = F k, L F k, R of the k th pair in the downlink framework.

Using a functional analytic approach, we give several surprising equivalent formulations for the equivalence problem, which reveal a deep connection to several fundamental problems in the integral geometry of the Grassmann manifold.

It is shown that the problem of finding higher-order strict minimizers with respect to a nonlinear function for the given problem reduces to that of finding strict minimizers in the ordinary sense for an equivalent problem.

This solution is valid for the equivalent heat transfer problem also.

In this paper, a SOR (Successive Over Relaxation) algorithm is designed for the equivalent bi-convex problem and its convergence is proved.

By applying mirror symmetry, mathematicians have translated this problem into an equivalent problem for the mirror Calabi Yau, which turns out to be easier to solve.

Investigating the uniqueness of a solution of a problem of integral geometry by reducing it to the equivalent inverse problem for a differential equation was first carried out in [30].