It has been known for almost 50 years that the discrete l_1 approximation problem can be solved effectively by linear programming. However, improved algorithms involve a step which can be interpreted as a line search, and which is not part of the standard LP solution procedures. l_1 provides the simplest example of a class of … Read more
Let f be a lower semi-continuous convex function in a Euclidean space, finite or infinite dimensional. The gradient differential inclusion is a generalized differential equation which requires that -x'(t) be in the subgradient of f at x. It is the continuous versions of the gradient descent method for minimizing f in case f is differentiable, … Read more
We present results on global and polynomial-time convergence of infeasible-interior-point methods for self-scaled conic programming, which includes linear and semidefinite programming. First, we establish global convergence for an algorithm using a wide neighborhood. Next, we prove polynomial complexity for the algorithm with a slightly narrower neighborhood. Both neighborhoods are related to the wide (minus infinity) … Read more
Recently, tools from algebraic geometry have been successfully applied to develop solution schemes for new classes of optimization problems. A central idea in these constructions is to express a polynomial that is positive on a given domain in terms of polynomials of higher degree so that its positivity is readily revealed. This resembles the “lifting” … Read more
Let $f$ be a continuous function on $\Rl^n$, and suppose $f$ is continuously differentiable on an open dense subset. Such functions arise in many applications, and very often minimizers are points at which $f$ is not differentiable. Of particular interest is the case where $f$ is not convex, and perhaps not even locally Lipschitz, but … Read more
The variational approach for electronic structure based on the two-body reduced density matrix is studied, incorporating two representability conditions beyond the previously used $P$, $Q$ and $G$ conditions. The additional conditions (called $T1$ and $T2$ here) are implicit in work of R.~M.~Erdahl [Int.\ J.\ Quantum Chem.\ {\bf13}, 697–718 (1978)] and extend the well-known three-index diagonal … Read more
We propose a pivotting procedure for a class of Second-Order Cone Programming (SOCP) having one second-order cone. We introduce a dictionary, basic variables, nonbasic variables, and other necessary notions to define a pivot for the class of SOCP. In a pivot, two-dimensional SOCP subproblems are solved to decide which variables should be entering to or … Read more
We propose a dynamic version of the bundle method to get approximate solutions to semidefinite programs with a nearly arbitrary number of linear inequalities. Our approach is based on Lagrangian duality, where the inequalities are dualized, and only a basic set of constraints is maintained explicitly. This leads to function evaluations requiring to solve a … Read more
We introduce a new masked spectral bound for the maximum-entropy sampling problem. This bound is a continuous generalization of the very effective spectral partition bound. Optimization of the masked spectral bound requires the minimization of a nonconvex, nondifferentiable function over a semidefiniteness constraint. We describe a nonlinear affine scaling algorithm to approximately minimize the bound. … Read more
An important measure of conditioning of a conic linear system is the size of the smallest structured perturbation making the system ill-posed. We show that this measure is unchanged if we restrict to perturbations of low rank. We thereby derive a broad generalization of the classical Eckart-Young result characterizing the distance to ill-posedness for a … Read more