Spectral optimization problem

The energy problem is accompanied by the dual spectral optimization problem,... 

This gap inequality follows from Lagrange s derivation of the dual spectral optimization problem, but there is a more direct proof that we now present. 

The search for optimal solutions to both the energy problem and the spectral optimization problem typically starts with matrices P and S that have a positive gap. Iterations are designed to move P so that the energy is decreased, and to move S so that the bottom eigenvalue is increased, and such motions cause the gap to narrow. It is important that there are semidefinite programs where this gap cannot shrink to zero, and we discuss such an example later. However, in our special case where we vary fe-matrices and Pauli matrices, as we have defined them, the gap shrinks to zero. This is an important result for both theoretical and practical reasons a proof is supplied below. 

The energy and spectral optimization problems are convex programs so when there are multiple solutions the solution sets form a convex set. The following corollary characterizes how these convex sets of solutions relate to solutions of the Euler equation. In the formulation of this corollary we use the notion of optimal gap Ao—the gap achieved by optimal P and S. The optimal gap is a characteristic of the energy problem, depending only on H and S. 

In this section we continue the discussion of the energy and spectral optimization problems,... 

To prove our main theorem, establishing that the existence of a solution of the spectral optimization problem is equivalent to the existence of a solution of the... 

If the subspace S contains a positive definite element S, neither the energy problem nor the spectral optimization problem has an optimal solution since there is no positive semidefinite matrix P satisfying both the conditions (P, = 0, (P, I)j = 1, the convex set Pq n 5 is empty, and the energy pro-... 

For the spectral optimization problem we must calculate the bottom eigenvalue of... 

Corollary 14 Assume that S is the Pauli subspace as defined in Section II. Then the energy minimization and spectral optimization problems have optimal solutions P, S, and PQ = O, where Q = H + S — /lo(H + S)I. 

Proof. By property R5 listed at the end of Section II, the elements of the Pauli subspace S are traceless, from which we infer by Theorems 12 and 13 that the energy problem and the spectral optimization problem have optimal solutions. By Theorem 10 these solutions are characterized by the Euler equation PQ = 0. ... 

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