This paper proposes a primal-dual interior-point filter method for nonlinear semidefinite programming, which is the first multidimensional (three-dimensional) filter methods for interior-point methods, and of course for constrained optimization. A freshly new definition of filter entries is proposed, which is greatly different from those in all the current filter methods. A mixed norm is used … Read more
Semidefinite Programming (SDP) may be seen as a generalization of Linear Programming (LP). In particular, one may extend interior point algorithms for LP to SDP, but it has proven much more difficult to exploit structure in the SDP data during computation. We survey three types of special structure in SDP data: 1) a common `chordal’ … Read more
We propose the DISCO algorithm for graph realization in $\real^d$, given sparse and noisy short-range inter-vertex distances as inputs. Our divide-and-conquer algorithm works as follows. When a group has a sufficiently small number of vertices, the basis step is to form a graph realization by solving a semidefinite program. The recursive step is to break … Read more
Solvers for semidefinite programming (SDP) have evolved a great deal in the last decade, and their development continues. In order to further support and encourage this development, we present a new test set of SDP instances. These instances arise from recent applications of SDP in coding theory, computational geometry, graph theory and structural design. Most … Read more
We observe that in a simple one-dimensional polynomial optimization problem (POP), the `optimal’ values of semidefinite programming (SDP) relaxation problems reported by the standard SDP solvers converge to the optimal value of the POP, while the true optimal values of SDP relaxation problems are strictly and significantly less than that value. Some pieces of circumstantial … Read more
The elegant results for strong duality and strict complementarity for linear programming, \LP, can fail for cone programming over nonpolyhedral cones. One can have: unattained optimal values; nonzero duality gaps; and no primal-dual optimal pair that satisfies strict complementarity. This failure is tied to the nonclosure of sums of nonpolyhedral closed cones. We take a … Read more
Interior-point methods for semidefinite optimization have been studied intensively, due to their polynomial complexity and practical efficiency. Recently, the second author designed an efficient primal-dual infeasible interior-point algorithm with full Newton steps for linear optimization problems. In this paper we extend the algorithm to semidefinite optimization. The algorithm constructs strictly feasible iterates for a sequence … Read more
We provide a specific representation of convex polynomials nonnegative on a convex (not necessarily compact) basic closed semi-algebraic set $K$ of $R^n$. Namely, they belong to a specific subset of the quadratic module generated by the concave polynomials that define $K$. CitationTo appear in Archiv der MathematikArticleDownload View PDF
We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence, a highly desirable feature as convex problems are in principle easier to … Read more
In this paper, we extend the Ai-Zhang direction to the class of semidefinite optimization problems. We define a new wide neighborhood $\N(\tau_1,\tau_2,\eta)$ and, as usual, we utilize symmetric directions by scaling the Newton equation with special matrices. After defining the “positive part” and the “negative part” of a symmetric matrix, we solve the Newton equation … Read more