On Optimality Conditions for Nonlinear Conic Programming

Sequential optimality conditions play a major role in proving stronger global convergence results of numerical algorithms for nonlinear programming. Several extensions are described in conic contexts, in which many open questions have arisen. In this paper, we present new sequential optimality conditions in the context of a general nonlinear conic framework, which explains and improves several known results for specific cases, such as semidefinite programming, second-order cone programming, and nonlinear programming. In particular, we show that feasible limit points of sequences generated by the augmented Lagrangian method satisfy the so-called approximate gradient projection optimality condition and, under an additional smoothness assumption, the so-called complementary approximate Karush–Kuhn–Tucker condition. The first result was unknown even for nonlinear programming, and the second one was unknown, for instance, for semidefinite programming.

Bregman primal–dual first-order method and application to sparse semidefinite programming

We present a new variant of the Chambolle–Pock primal–dual algorithm with Bregman distances, analyze its convergence, and apply it to the centering problem in sparse semidefinite programming. The novelty in the method is a line search procedure for selecting suitable step sizes. The line search obviates the need for estimating the norm of the constraint matrix and the strong convexity constant of the Bregman kernel. As an application, we discuss the centering problem in large-scale semidefinite programming with sparse coefficient matrices. The logarithmic barrier function for the cone of positive semidefinite completable sparse matrices is used as the distance-generating kernel. For this distance, the complexity of evaluating the Bregman proximal operator is shown to be roughly proportional to the cost of a sparse Cholesky factorization. This is much cheaper than the standard proximal operator with Euclidean distances, which requires an eigenvalue decomposition.

Efficient Semidefinite Programming with Approximate ADMM

Tenfold improvements in computation speed can be brought to the alternating direction method of multipliers (ADMM) for Semidefinite Programming with virtually no decrease in robustness and provable convergence simply by projecting approximately to the Semidefinite cone. Instead of computing the projections via “exact” eigendecompositions that scale cubically with the matrix size and cannot be warm-started, we suggest using state-of-the-art factorization-free, approximate eigensolvers, thus achieving almost quadratic scaling and the crucial ability of warm-starting. Using a recent result from Goulart et al. (Linear Algebra Appl 594:177–192, 2020. https://doi.org/10.1016/j.laa.2020.02.014), we are able to circumvent the numerical instability of the eigendecomposition and thus maintain tight control on the projection accuracy. This in turn guarantees convergence, either to a solution or a certificate of infeasibility, of the ADMM algorithm. To achieve this, we extend recent results from Banjac et al. (J Optim Theory Appl 183(2):490–519, 2019. https://doi.org/10.1007/s10957-019-01575-y) to prove that reliable infeasibility detection can be performed with ADMM even in the presence of approximation errors. In all of the considered problems of SDPLIB that “exact” ADMM can solve in a few thousand iterations, our approach brings a significant, up to 20x, speedup without a noticeable increase in ADMM’s iterations.

Perturbation Propagation Models for Underwater Sensor Localisation using Semidefinite Programming

Real time Underwater sensor networks (UWSNs) suffer from localisation issues due to a dearth of incorporation of different geometric scenarios in UWSN scenarios. To address these issues, this paper visualises three specific scenarios of perturbation. First, small sized and large numbered particles of perturbance moving in a tangential motion to the sensor nodes; second, a single numbered and large-sized particle moving in a rectilinear motion by displacing the sensor nodes into sideward and forward direction, and third, a radially outward propagating perturbance to observe the influenced sensor nodes as the perturbance moves outwards. A novel target localisation and tracking is facilitated by including marine vehicle navigation as a source of perturbation. Using semidefinite programming, the proposed perturbation models minimise localisation errors, thereby enhancing physical security of underwater sensor nodes. By leveraging the spin, cleaving motion and radial cast-away behaviour of underwater sensor nodes, the results confirm that the proposed propagation models can be conveniently applied to real time target detection and estimation of underwater target nodes.

SDP-Based Bounds for the Quadratic Cycle Cover Problem via Cutting-Plane Augmented Lagrangian Methods and Reinforcement Learning

We study the quadratic cycle cover problem (QCCP), which aims to find a node-disjoint cycle cover in a directed graph with minimum interaction cost between successive arcs. We derive several semidefinite programming (SDP) relaxations and use facial reduction to make these strictly feasible. We investigate a nontrivial relationship between the transformation matrix used in the reduction and the structure of the graph, which is exploited in an efficient algorithm that constructs this matrix for any instance of the problem. To solve our relaxations, we propose an algorithm that incorporates an augmented Lagrangian method into a cutting-plane framework by utilizing Dykstra’s projection algorithm. Our algorithm is suitable for solving SDP relaxations with a large number of cutting-planes. Computational results show that our SDP bounds and efficient cutting-plane algorithm outperform other QCCP bounding approaches from the literature. Finally, we provide several SDP-based upper bounding techniques, among which is a sequential Q-learning method that exploits a solution of our SDP relaxation within a reinforcement learning environment. Summary of Contribution: The quadratic cycle cover problem (QCCP) is the problem of finding a set of node-disjoint cycles covering all the nodes in a graph such that the total interaction cost between successive arcs is minimized. The QCCP has applications in many fields, among which are robotics, transportation, energy distribution networks, and automatic inspection. Besides this, the problem has a high theoretical relevance because of its close connection to the quadratic traveling salesman problem (QTSP). The QTSP has several applications, for example, in bioinformatics, and is considered to be among the most difficult combinatorial optimization problems nowadays. After removing the subtour elimination constraints, the QTSP boils down to the QCCP. Hence, an in-depth study of the QCCP also contributes to the construction of strong bounds for the QTSP. In this paper, we study the application of semidefinite programming (SDP) to obtain strong bounds for the QCCP. Our strongest SDP relaxation is very hard to solve by any SDP solver because of the large number of involved cutting-planes. Because of that, we propose a new approach in which an augmented Lagrangian method is incorporated into a cutting-plane framework by utilizing Dykstra’s projection algorithm. We emphasize an efficient implementation of the method and perform an extensive computational study. This study shows that our method is able to handle a large number of cuts and that the resulting bounds are currently the best QCCP bounds in the literature. We also introduce several upper bounding techniques, among which is a distributed reinforcement learning algorithm that exploits our SDP relaxations.

A Relaxed Interior Point Method for Low-Rank Semidefinite Programming Problems with Applications to Matrix Completion

A new relaxed variant of interior point method for low-rank semidefinite programming problems is proposed in this paper. The method is a step outside of the usual interior point framework. In anticipation to converging to a low-rank primal solution, a special nearly low-rank form of all primal iterates is imposed. To accommodate such a (restrictive) structure, the first order optimality conditions have to be relaxed and are therefore approximated by solving an auxiliary least-squares problem. The relaxed interior point framework opens numerous possibilities how primal and dual approximated Newton directions can be computed. In particular, it admits the application of both the first- and the second-order methods in this context. The convergence of the method is established. A prototype implementation is discussed and encouraging preliminary computational results are reported for solving the SDP-reformulation of matrix-completion problems.