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Operator convex functions defined on the positive half-line play a prominent role in the theory of quantum information, where they are used to define quantum $f$-divergences. Such functions admit integral representations in terms of…
Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in $H^k(\Omega)$, and optimizing functionals arising from some…
Many problems of theoretical and practical interest involve finding a convex or concave function. For instance, optimization problems such as finding the projection on the convex functions in $H^k(\Omega)$, or some problems in economics. In…
Optimization of convex functions subject to eigenvalue constraints is intriguing because of peculiar analytical properties of eigenvalues, and is of practical interest because of wide range of applications in fields such as structural…
We propose a general method for optimization with semi-infinite constraints that involve a linear combination of functions, focusing on the case of the exponential function. Each function is lower and upper bounded on sub-intervals by…
We consider positive, integral-preserving linear operators acting on $L^1$ space, known as stochastic operators or Markov operators. We show that, on finite-dimensional spaces, any stochastic operator can be approximated by a sequence of…
We consider the problem of minimizing a linear function over an affine section of the cone of positive semidefinite matrices, with the additional constraint that the feasible matrix has prescribed rank. When the rank constraint is active,…
Semidefinite programs are convex optimisation problems involving a linear objective function and a domain of positive semidefinite matrices. Over the last two decades, they have become an indispensable tool in quantum information science.…
We describe several algorithms for matrix completion and matrix approximation when only some of its entries are known. The approximation constraint can be any whose approximated solution is known for the full matrix. For low rank…
A new algorithm to approximate Hermitian matrices by positive semidefinite Hermitian matrices based on modified Cholesky decompositions is presented. In contrast to existing algorithms, this algorithm allows to specify bounds on the…
This short but self-contained survey presents a number of elegant matrix/operator inequalities for general convex or concave functions, obtained with a unitary orbit technique. Jensen, sub or super-additivity type inequalities are…
The completely positive maps, a generalization of the nonnegative matrices, are a well-studied class of maps from $n\times n$ matrices to $m\times m$ matrices. The existence of the operator analogues of doubly stochastic scalings of…
The approximate representation of operators by finite matrices is analysed in terms of accuracy and convergence. The identity operator, for example, can be reconstructed using a basis of harmonic oscillator states leading to a narrow peak…
Given a set of matrices, modeled as samples of a matrix-valued function, we suggest a method to approximate the underline function using a product approximation operator. This operator extends known approximation methods by exploiting the…
We address the problem of minimizing a convex function over the space of large matrices with low rank. While this optimization problem is hard in general, we propose an efficient greedy algorithm and derive its formal approximation…
This note provides another proof for the {\em convexity} ({\em strict convexity}) of $\log \det ( I + KX^{-1} )$ over the positive definite cone for any given positive semidefinite matrix $K \succeq 0$ (positive definite matrix $K \succ 0$)…
We describe an elementary algorithm to build convex inner approximations of nonconvex sets. Both input and output sets are basic semialgebraic sets given as lists of defining multivariate polynomials. Even though no optimality guarantees…
We present a hybrid algorithm for optimizing a convex, smooth function over the cone of positive semidefinite matrices. Our algorithm converges to the global optimal solution and can be used to solve general large-scale semidefinite…
We consider the minimization or maximization of the $J$th largest eigenvalue of an analytic and Hermitian matrix-valued function, and build on Mengi et al. (2014, SIAM J. Matrix Anal. Appl., 35, 699-724). This work addresses the setting…
The affine inverse eigenvalue problem consists of identifying a real symmetric matrix with a prescribed set of eigenvalues in an affine space. Due to its ubiquity in applications, various instances of the problem have been widely studied in…