Matrix optimization has proven to be extremely important in applications ranging across the human experience. Two sides exist to matrices: the coordinate side and the spectral side. In this talk, we shall present our latest work expanding the ability to solve optimization problems with general constraints on both the eigenvalues or singular values of a matrix and its coordinates.

In this talk, we consider the cone of linear mappings whose associated quadratic forms are nonnegative over symmetric cones, referred to as copositive cones over symmetric cones. We show that copositive cones are never facially exposed when the underlying symmetric cone has dimension at least 2. To achieve this, we explicitly exhibit a non-exposed extreme ray. Our result extends the known fact that the cone of copositive matrices in the conventional sense is generally not facially exposed.

Optimization over the quantum relative entropy (QRE) cone has many applications in quantum information processing, such as calculating the key rates for quantum key distribution (QKD) protocols. Recently, a new version of our convex optimization software package Domain-Driven Solver (DDS) was released with modified numerical approaches for solving QRE programming problems, potentially combined with many other convex function/set constraints. In this talk, we propose a preprocessing and two-phase approach for QRE programming to reduce the size and improve the conditioning of a given instance. Phase-I of this two-phase approach is a well-behaved convex optimization problem, such as minimizing a self-concordant barrier or Semi-Definite Programming. The reformulated QRE programming problem can be solved more efficiently and faster by DDS or other optimization algorithms. We finish the talk by presenting numerical results of using DDS to solve many classes of problems, including calculating the QKD rate for different protocols.

Given closed convex sets $C_i$, $i=1,\ldots,\ell$, and some nonzero linear maps $A_i$, $i = 1,\ldots,\ell$,
of suitable dimensions, the multi-set split feasibility problem aims at finding a point in $\bigcap_{i=1}^\ell A_i^{-1}C_i$ based on computing projections onto $C_i$ and multiplications by $A_i$ and $A_i^T$. In this paper, we consider the associated best approximation problem, i.e., the problem of computing projections onto $\bigcap_{i=1}^\ell A_i^{-1}C_i$; we refer to this problem as the best approximation problem in multi-set split feasibility settings (BA-MSF). We adapt the Dykstra's projection algorithm, which is classical for solving the BA-MSF in the special case when all $A_i = I$, to solve the general BA-MSF. Our Dykstra-type projection algorithm is derived by applying (proximal) coordinate gradient descent to the Lagrange dual problem, and it only requires computing projections onto $C_i$ and multiplications by $A_i$ and $A_i^T$ in each iteration. Under a standard relative interior condition and a genericity assumption on the point we need to project, we show that the dual objective satisfies the Kurdyka-{\L}ojasiewicz property with an explicitly computable exponent on a neighborhood of the (typically unbounded) dual solution set when each $C_i$ is $C^{1,\alpha}$-cone reducible for some $\alpha\in (0,1]$: this class of sets covers the class of $C^2$-cone reducible sets, which include all polyhedrons, second-order cone, and the cone of positive semidefinite matrices as special cases. Using this, explicit convergence rate (linear or sublinear) of the sequence generated by the Dykstra-type projection algorithm is derived. Concrete examples are constructed to illustrate the necessity of some of our assumptions.