We introduce PyCFTBoot, a wrapper designed to reduce the barrier to entry in conformal bootstrap calculations that require semidefinite programming. Symengine and SDPB are used for the most intensive symbolic and numerical steps respectively. After reviewing the built-in algorithms for conformal blocks, we explain how to use the code through a number of examples that verify past results. As an application, we show that the multi-correlator bootstrap still appears to single out the Wilson-Fisher fixed points as special theories in dimensions between 3 and 4 despite the recent proof that they violate unitarity.

In this paper, we show that the direct semidefinite programming (SDP) bound for thenonconvex quadratic optimization problem over ℓ1 unit ball(QPL1) is equivalent to the optimal d.c. (difference between convex) bound for thestandard quadratic programming reformulation of QPL1. Then we disprove a conjecture aboutthe tightness of the direct SDP bound. Finally, as an extension of QPL1, we study therelaxation problem of the sparse principal component analysis, denoted by QPL2L1. We showthat the existing direct SDP bound for QPL2L1 is equivalent to the doubly nonnegativerelaxation for variable-splitting reformulation of QPL2L1.

In this paper we analyze a known relaxation for the Sparsest Cut problem based on positive semidefinite constraints, and we present a branch and bound algorithm and heuristics based on this relaxation. The relaxed formulation and the algorithms were tested on small and moderate sized instances. It leads to values very close to the optimum solution values. The exact algorithm could obtain solutions for small and moderate sized instances, and the best heuristics obtained optimum or near optimum solutions for all testedinstances. The semidefinite relaxation gives a lower bound $\frac{C}{W}$ and each heuristic produces a cut S with a ratio $\frac{c_S}{w_S}$, where either cS is at most a factor of C or wS is at least a factor of W. We solved the semidefinite relaxation using a semi-infinite cut generation with a commercial linear programming package adapted to the sparsest cut problem. We showed that the proposed strategy leads to a better performance compared to the use of a known semidefinite programming solver.

There has been much interest recently in smoothing methods for solving semidefinite programming (SDP). In this paper, based on the equivalent transformation for the optimality conditions of SDP, we present a predictor–corrector smoothing Newton algorithm for SDP. Issues such as the existence of Newton directions, boundedness of iterates, global convergence, and local superlinear convergence of our algorithm are studied under suitable assumptions.

The goal of this paper is to study the so-called worst-case or robustoptimal design problem for minimal compliance. In the context of linear elasticity we seek an optimal shape which minimizes the largest, or worst,compliance when the loads are subject to some unknown perturbations.We first prove that, for a fixed shape, there exists indeed a worst perturbation (possibly non unique) that we characterize as the maximizer of a nonlinear energy. We also propose a stable algorithm tocompute it. Then, in the framework of Hadamard method, wecompute the directional shape derivative of this criterion which isused in a numerical algorithm, based on the level set method,to find optimal shapes that minimize the worst-case compliance.Since this criterion is usually merely directionally differentiable,we introduce a semidefinite programming approach toselect the best descent direction at each step of agradient method. Numerical examples are given in 2-d and 3-d.


The ground-state energy and properties of any many-electron atom ormolecule may be rigorously computed by variationally computing thetwo-electron reduced density matrix rather than the many-electronwavefunction. While early attempts fifty years ago to compute theground-state 2-RDM directly were stymied because the 2-RDM must beconstrained to represent an N-electron wavefunction, recentadvances in theory and optimization have made direct computation ofthe 2-RDM possible. The constraints in the variational calculationof the 2-RDM require a special optimization known as a semidefiniteprogramming. Development of first-order semidefinite programmingfor the 2-RDM method has reduced the computational costs of thecalculation by orders of magnitude [Mazziotti, Phys. Rev. Lett.93 (2004) 213001]. The variational 2-RDM approach is effective atcapturing multi-reference correlation effects that are especiallyimportant at non-equilibrium molecular geometries. Recent work on2-RDM methods will be reviewed and illustrated with particularemphasis on the importance of advances in large-scale semidefiniteprogramming.


This paper presents a feasible primal algorithm for linear semidefinite programming. The algorithm starts with a strictly feasible solution, but in case where no such a solution is known, an application of the algorithm to an associate problem allows to obtain one. Finally, we present some numerical experiments which show that the algorithm works properly.

In this paper we study the well definedness of the central path associated to agiven nonlinear (convex) semidefinite programming problem. Under standard assumptions,we establish that the existence of the central path is equivalent to the nonemptiness andboundedness of the optimal set. Other equivalent conditions are given, such as the existenceof a strictly dual feasible point or the existence of a single central point.The monotonicbehavior of the logarithmic barrier and the objective function along the trajectory is alsodiscussed. Finally, the existence and optimality of cluster points are established.

This paper is devoted to the computation of distance to set, called S, defined by polynomial equations. First we consider the case of quadratic systems. Then, application of results stated for quadratic systems to the quadratic equivalent of polynomial systems (see [5]), allows us to compute distance to semi-algebraic sets. Problem of computing distance can be viewed as non convex minimization problem: $ d(u,S) = \inf_{x \in S} \| x-u\|^2$, where u is in $\mathcal{R}^{n}$. To have, at least, lower approximation of distance, we consider the dual bound m(u) associated with the dual problem and give sufficient conditions to guarantee m(u) = d(u,S). The second part deal with numerical computation of m(u) using an interior point method in semidefinite programming. Last, various examples, namely from chemistry and robotic, are given.