We propose a manifold optimization approach to solve linear semidefinite programs (SDP) with low-rank solutions. This approach incorporates the augmented Lagrangian method and the Burer-Monteiro factorization, and features the adaptive strategies for updating the factorization size and the penalty parameter. We prove that the present algorithm can solve SDPs to global optimality, despite of the … Read more
In this paper, we introduce a new effective matrix adaptation evolution strategy (MADFO) for noisy derivative-free optimization problems. Like every MAES solver, MADFO consists of three phases: mutation, selection and recombination. MADFO improves the mutation phase by generating good step sizes, neither too small nor too large, that increase the probability of selecting mutation points … Read more
This thesis studies derivative-free optimization (DFO), particularly model-based methods and software. These methods are motivated by optimization problems for which it is impossible or prohibitively expensive to access the first-order information of the objective function and possibly the constraint functions. In particular, this thesis presents PDFO, a package we develop to provide both MATLAB and Python … Read more
The late Professor M. J. D. Powell devised five trust-region derivative-free optimization methods, namely COBYLA, UOBYQA, NEWUOA, BOBYQA, and LINCOA. He also carefully implemented them into publicly available solvers, which are renowned for their robustness and efficiency. However, the solvers were implemented in Fortran 77 and hence may not be easily accessible to some users. … Read more
Nonnegativity certificates can be used to obtain tight dual bounds for polynomial optimization problems. Hierarchies of certificate-based relaxations ensure convergence to the global optimum, but higher levels of such hierarchies can become very computationally expensive, and the well-known sums of squares hierarchies scale poorly with the degree of the polynomials. This has motivated research into … Read more
Symmetry in optimization has been known to wreak havoc in optimization algorithms. Often, some of the hardest instances are highly symmetric. This is not the case in linear programming, as symmetry allows one to reduce the size of the problem, possibly dramatically, while still maintaining the same optimal objective value. This is done by aggregating … Read more
For over ten years, the constraint integer programming framework SCIP has been extended by capabilities for the solution of convex and nonconvex mixed-integer nonlinear programs (MINLPs). With the recently published version 8.0, these capabilities have been largely reworked and extended. This paper discusses the motivations for recent changes and provides an overview of features that … Read more
Optimization over the embedded submanifold defined by constraints $c(x) = 0$ has attracted much interest over the past few decades due to its wide applications in various areas, including computer vision, signal processing, numerical linear algebra, and deep learning. Plenty of related optimization packages have been developed based on Riemannian optimization approaches, which rely on … Read more
This paper discusses a composite algorithm for bound constrained noisy derivative-free optimization problems with integer variables. This algorithm is an integer variant of the matrix adaptation evolution strategy. An integer derivative-free line search strategy along affine scaling matrix directions is used to generate candidate points. Each affine scaling matrix direction is a product of the … Read more
An essential component in modern solvers for mixed-integer (linear) programs (MIPs) is the separation of additional inequalities (cutting planes) to tighten the linear programming relaxation. Various algorithmic decisions are necessary when integrating cutting plane methods into a branch-and-bound (B&B) solver as there is always the trade-off between the efficiency of the cuts and their costs, … Read more