The proximal point mapping is the basis of many optimization techniques for convex functions. By means of variational analysis, the concept of proximal mapping was recently extended to nonconvex functions that are prox-regular and prox-bounded. In such a setting, the proximal point mapping is locally Lipschitz continuous and its set of fixed points coincide with … Read more
We describe a Jordan-algebraic version of results related to convexity of images of quadratic mappings as well as related results on exactness of symmetric relaxations of certain classes of nonconvex optimization problems. The exactness of relaxations is proved based on rank estimates. Our approach provides a unifying viewpoint on a large number of classical results … Read more
Suppose x and s lie in the interiors of a cone K and its dual K^* respectively. We seek dual ellipsoidal norms such that the product of the radii of the largest inscribed balls centered at x and s and incribed in K and K^* respectively is maximized. Here the balls are defined using the … Read more
We present a full-Newton step infeasible interior-point algorithm. It is shown that at most $O(n)$ (inner) iterations suffice to reduce the duality gap and the residuals by the factor $\frac1{e}$. The bound coincides with the best known bound for infeasible interior-point algorithms. It is conjectured that further investigation will improve the above bound to $O(\sqrt{n})$. … Read more
We analyze the problem of finding a point strictly interior to a bounded, fully dimensional set from a finite dimensional Hilbert space. We generalize the results obtained for the LP, SDP and SOCP cases. The cuts added by our algorithm are central and conic. In our analysis, we find an upper bound for the number … Read more
Primal-dual Interior-Point Methods (IPMs) have shown their ability in solving large classes of optimization problems efficiently. Feasible IPMs require a strictly feasible starting point to generate the iterates that converge to an optimal solution. The self-dual embedding model provides an elegant solution to this problem with the cost of slightly increasing the size of the … Read more
For a locally optimal solution to the nonlinear semidefinite programming problem, under Robinson’s constraint qualification, the following conditions are proved to be equivalent: the strong second order sufficient condition and constraint nondegeneracy; the nonsingularity of Clarke’s Jacobian of the Karush-Kuhn-Tucker system; the strong regularity of the Karush-Kuhn-Tucker point; and others. CitationTechnical Report, Department of Mathematics, … Read more
The inverse quadratic eigenvalue problem (IQEP) arises in the field of structural dynamics. It aims to find three symmetric matrices, known as the mass, the damping and the stiffness matrices, respectively such that they are closest to the given analytical matrices and satisfy the measured data. The difficulty of this problem lies in the fact … Read more
We investigate ill-posed semidefinite programming problems for which Slater’s constraint qualifications fail, and propose a new reliable termination criterium dealing with such problems. This criterium is scale-independent and provides verified forward error bounds for the true optimal value, where all rounding errors due to floating point arithmetic are taken into account. It is based on … Read more
We establish new lower bounds on the distance between two points of a minimum energy configuration of $N$ points in $\mathbb{R}^3$ interacting according to a pairwise potential function. For the Lennard-Jones case, this bound is 0.67985 (and 0.7633 in the planar case). A similar argument yields an estimate for the minimal distance in Morse clusters, … Read more