We propose and study a weakly convergent variant of the forward--backward algorithm for solving structured monotone inclusion problems. Our algorithm features a per-iteration deviation vector which provides additional degrees of freedom. The only requirement on the deviation vector to guarantee convergence is that its norm is bounded by a quantity that can be computed online. This approach provides great flexibility and opens up for the design of new and improved forward--backward-based algorithms, while retaining global convergence guarantees. These guarantees include linear convergence of our method under a metric subregularity assumption without the need to adapt the algorithm parameters. Choosing suitable monotone operators allows for incorporating deviations into other algorithms, such as Chambolle--Pock and Krasnoselsky--Mann iterations. We propose a novel inertial primal--dual algorithm by selecting the deviations along a momentum direction and deciding their size using the norm condition. Numerical experiments demonstrate our convergence claims and show that even this simple choice of deviation vector can improve the performance, compared, e.g., to the standard Chambolle--Pock algorithm.
We propose a special facial reduction algorithm (FRA) for the Shor semidefinite programming (SDP) relaxation of the quadratically constrained quadratic program (QCQP).Under the mild assumption, our special FRA only requires solving a linear programming problem instead of a semidefinite program.In particular, when applied to the binary quadratic program, the proposed special FRA needs fewer assumptions.From a computational perspective, this result improves the scalability and stability of the SDP approach for QCQP problems.In addition, we also discover a new class of semidefinite programs whose singularity degree can be computed easily.This new class complements the limited examples of singularity degrees in the literature.As a by-product, our special FRA can be used to upper bound the dimension of any set defined by quadratic inequalities.
Quasi-Newton method is one of the most popular methods for solving unconstrained single and multiobjective optimization problems.In a quasi-Newton method, the search direction is computed based on a quadratic model of the objective function, where some approximations replace the true Hessian at each iteration.Several Hessian approximation schemes with an adequate line search technique provided higher-order accuracies in approximating the curvature and made the methods more effective in terms of convergence to solution.Considering the aforementioned reasons, we write a survey on quasi-Newton methods for multiobjective optimization problems.We discuss the development of all the variants of the quasi-Newton method for multiobjective optimization problems, along with some of the advantages and disadvantages of the existing methods.We give a brief discussion about the line search for these variants too.Two cases have been considered for BFGS, Huang BFGS, and self-scaling BFGS multiobjective versions of quasi-Newton methods: one is in the presence of the Armijo line search, and the other is in the absence of any line search.Subsequently, a nonmonotone line search version is also explained for multiobjective optimization problems.Commentary is given on the convergence properties of these methods.The rate of convergence of all these methods is highlighted.To prove the convergence of every method, it is reported that every sequence of points generated from the method converges to a critical point of the multiobjective optimization problem under some mild assumptions.Based on the available numerical data, we provide an unbiased opinion on the effectiveness of quasi-Newton methods for multiobjective optimization problems.
We present an algorithm for computing the minimum-rank positive semidefinite completion of a sparse matrix with a chordal sparsity pattern.This problem is tractable, in contrast to the minimumrank positive semidefinite completion problem for general sparsity patterns.We also present a similar algorithm for the Euclidean distance matrix completion with minimum embedding dimension.The two algorithms use efficient recursions over a clique tree associated with the chordal sparsity pattern.As an application, we use the minimum-rank completion method as a rounding technique to convert the solution of a sparse semidefinite optimization problem with non-unique solutions to an optimal solution of lower rank.In experiments with semidefinite relaxations of optimal power flow problems, the minimum-rank completion often results in solutions of lower rank than the solutions computed by interior-point solvers.
The paper is devoted to the calmness from below/from above for the optimal value function ϕ of parametric optimization problems, where we are specifically interested in perturbed semi-infinite programs.A main intention is to revisit classical results and to derive refinements of them.In particular, we show in the context of semi-infinite optimization that calmness from below for ϕ holds under quasiconvexity of the data functions and compactness of the solution set, which extends results on the lower semicontinuity of ϕ.Illustrative examples are given, which demonstrate the significance of the imposed assumptions even in the case of linear and quadratic programs.
