A new mathematical approach to finding global solutions of the magnetic structure determination problem

Determination of magnetic structure is an important analytical procedure utilized in various fields ranging from fundamental condensed-matter physics and chemistry to advanced manufacturing. It is typically performed using a neutron diffraction technique; however, finding global solutions of the magnetic structure optimization problem represents a significant challenge. Generally, it is not possible to mathematically prove that the obtained magnetic structure is a truly global solution and that no solution exists when no acceptable structure is found. In this study, the global optimization technique called semidefinite relaxation of quadratic optimization, which has attracted much interest in the field of applied mathematics, is proposed to use as a new analytical method for the determination of magnetic structure, followed by the application of polarized neutron diffraction data. This mathematical approach allows avoiding spurious local solutions, decreasing the amount of time required to find a tentative solution and finding multiple solutions when they exist.


I. Details of Mathematical Formulation A. General descriptions of the SDP and SDR techniques
The following material is based on the known theorems of the field of convex optimization 24 . A typical SDP optimization problem can be represented by one of the following forms: When the parameters Ak, C, and 8 are common for forms (i) and (ii), either (i) or (ii) is called a primal problem, while the other one is called a dual problem. In this study, problem (A1) is considered the primal problem for clarity. According to the duality theorem, it is possible to determine whether the obtained solutions X= X (opt) and Y= Y (opt) are truly the global optimums by checking if the duality gap ∆ = (i) (opt) − (ii) (opt) between the optimum values (i) (opt) and (ii) (opt) is close to zero, or more precisely, as small as the square root of the machine epsilon.
The weak and strong duality theorems assert that (i) (opt) ≥ (ii) (opt) and that if the solutions X > 0 and Y > 0 satisfying all constraints in forms (i) and (ii), respectively, exist, the SDP problems (A1) and (A2) have equal optimum values. In this study, the existence of such strictly feasible solutions can be easily verified. Further, the complementary slackness theorem assures that Page 2 of 8 X (opt) Y (opt) = Y (opt) X (opt) = 0; hence, the solutions X (opt) and Y (opt) share the same system of eigenvectors. The methodology of interior point methods also ensures that the numerically obtained X (opt) and Y (opt) parameters satisfy the non-degeneracy condition rank X (opt) + rank Y (opt) = n. The SDR technique extends the global-convergence property of SDP to nonconvex optimization problems including the described case; in a broad sense, SDR is a method for approximating the global optimum value. For example, the quadratic programming problem ( where 8 and are the symmetric matrices, and 8 are the real numbers. The last two conditions imply the existence of the vector x = (x1, …, xn) T with X = xx T . If X is the global optimum of problem (A4), the parameters x1, …, xn represent the global solution of problem (3), and vice versa. If the rank-1 condition is removed, problem (A4) becomes an SDP one. The word relaxation generally indicates the techniques utilized to obtain information about the global optimums by removing some constraints in order to reduce the degree of computational complexity. More narrowly, semidefinite relaxation (SDR) denotes the techniques transforming the nonconvex optimization problem to an SDP one (for example, by removing the rank-1 constraints). After relaxation, the global optimum can be easily determined by the SDP solver. The global optimum value f (gl) is never greater than that of the original problem (A4), because the permitted range of X is simply increased by the relaxation. Thus, the information about the original problem (3) can be obtained using the relaxation of problem (A4). However, the global optimum denoted as X (opt) is sometimes of rank > 1 after relaxation. Furthermore, if multiple global optimums of rank-1 satisfying problem (A4) exist, all the rank-1 solutions are superimposed in X (opt) as follows: (nop) = M + ⋯ + r + M + ⋯ + r , V : rank-1 global optimum for problem (A4) V : rank > 1 global optimum after the relaxation of problem (A4), which cannot be represented as a sum of other global optimums of smaller ranks The superposition principle described above is based on the two following properties: ) are optimal ones, any combination of (λ1X1 + λ2X2, λ1yk + λ2zk, λ1 8 ] + λ2 8 ] , λ1 8 + λ2 8 ) that satisfies the conditions λ1+ λ2 = 1, λ1 > 0, and λ2 > 0 is also optimal. (Strict complementary slackness property): the numerical solution X (opt) and dual optimum Y (opt) obtained by the SDP solver satisfy the condition rank X (opt) + rank Y (opt) = n, owing to the methodology of interior point techniques. In the case of problem (A4), the optimum value f (gl) cannot be significantly reduced by relaxation since all 8 ] + 8 values are constrained to be non-negative. Therefore, if the utilized structural model is an appropriate one, the optimum value f (gl) /( − ƒ ) can be as small as "2⁄ . SDR can prove the uniqueness of the obtained solution if the corresponding X (opt) is of rank 1; otherwise, the obtained result represents a set of algebraic equations that define all global optimums of problem (A4). Hence, the multiple global optimums can be computed by solving these equations.

C. The standard SDP form of the relaxation problem (A4)
After taking the relaxation process into account, the following SDP form can be derived from problem (A4) Below is the standard form (problem (A1)) obtained for the SDP problem described above: where , › ™ , 8 (1 ≤ ≤ ), oe]V (1 ≤ ≤ ) are the following symmetric matrices with two n-by-n and one 2N-by-2N diagonal blocks. Here, Zn,l is the n-by-n matrix whose (l, l) fields are equal to 1, and the others are filled with zeros. On is the n-by-n zero matrix.

D. Evaluation of the obtained solution
If the duality gap Δ is close to zero, the global convergence of the SDP method has been achieved. Hence, the following conclusions can be made: If the globally optimal value of is not small, the corresponding type of magnetic structure is mathematically not valid; and If the value is small enough, the global optimum solution of problem (2) is numerically obtained using the relations X (opt) = xx T (if the rank of X (opt) is 1), and Y (opt) x = 0 (otherwise). In the former case, the obtained x represents the unique solution. Y (opt) x = 0 is a consequence of the complementary slackness theorem.

E. Neutron scattering cross-sections
The neutron scattering cross-sections of nuclear and magnetic reflections can be expressed by the following formula 17 : where nuc denotes the nuclear scattering vector hnuc knuc lnuc, C is the experimental scale constant identical to that for Imag, = 1/(sin 2 sin ) is the Lorentz factor, 2 is the scattering angle, the summation of nuc includes multiplicity, fj is the scattering length of the nucleus of atom j, and rj is the atomic position. Further, where mag denotes the magnetic scattering vector hmag kmag lmag, ª is the gyromagnetic ratio of the neutron, r0 is the classical electron radius, the summation of mag includes multiplicity, and are the Cartesian directions, mj is the magnetic moment at rj, and FM, j is the magnetic form factor of atom j. Using these formulas and the standard least-squares method, it is possible to numerically determine the common C value and other structural parameters from Inuc, while Imag produces the coefficient matrix S as a quadratic function of W,-. Thus, magnetic structure analysis can be performed as problem (2).

II. Separation of Imag and Inuc values
In this experiment, the polarization vector P is oriented parallel to the scattering vector Q to eliminate the complex nuclear-magnetic interference term from the neutron scattering crosssection 17 . As a result, the following equations are obtained, Hence, where R = N+/N− is the flipping ratio, P = (N+ − N−)/(N+ + N−) is the polarization value, and rmix = N−/(N+ + N−) is the mixing rate of minority-spin neutrons in a beam. The value of rmix is obtained at a paramagnetic temperature of 40 K (Imag = 0). The additional neutron scattering data are shown in Fig. S1, and the corresponding magnetic and nuclear intensities are listed in Tables S1 and S2, respectively.