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.

The Development of Relational Reasoning: An Eyetracking Analysis of Strategy Use and Adaptation in Children and Adults Performing Matrix Completion

Relational reasoning is a key component of fluid intelligence and important predictor of academic achievement. Relational reasoning is commonly assessed using matrix completion tasks, in which participants see an incomplete matrix of items that vary on different dimensions and must select an item that best completes the matrix based on the relations among items. Performance on such assessments increases dramatically across childhood into adulthood. However, despite widespread use, little is known about the strategies associated with good or poor matrix completion performance in childhood. This study examined the strategies children and adults use to solve matrix completion problems, how those strategies change with age, and whether children and adults adapt strategies to difficulty. We investigated 6- and 9-year-old children and adults performing matrix completion while undergoing eyetracking to infer strategy use. Across all ages, scanning across matrix rows and columns predicted good performance, and higher rates of consulting potential answers predicted poor performance, indicating that optimal matrix completion strategies are similar across development. Indices of good strategy use increased across childhood. As problems increased in difficulty, children and adults increased their scanning of matrix rows and columns, and adults and 9-year-olds also qualitatively shifted strategies to rely more on consulting potential answers. Adapting strategies to matrix difficulty, particularly increased scanning of rows and columns, was associated with good overall performance in both children and adults. These findings underscore the importance of both spontaneous and adaptive strategy use in individual differences in relational reasoning and its development.

Distinguishing an accomplice who removes obstacles to committing a crime from a co-executor

The article researches the practice of delimiting an accomplice from a co-executor. In cases where an accomplice intellectually or physically assists in the commission of a crime in the future or promises to provide such assistance after its completion, problems with the qualifications of his actions, as a rule, do not arise. A particular difficulty represents the legal assessment of the elimination of obstacles by the accomplice of the encroachment by the “main” performer. The author sets himself the goal of a scientifically grounded definition of the type of accomplice in a crime, which contributes to its commission by eliminating obstacles in the process of fulfilling the objective side. To achieve this goal, the normative and formal legal approaches were used, the decisions of the Supreme Court of the Russian Federation and the doctrinal provisions of the criminal were studied were studied. In the article, the author reasonably proposes that a person who removes obstacles to the implementation of the method of crime during the execution of its objective side be recognized as a co-executor. When deciding on the content of the objective side, one should issue not only from the abstract features of the corresponding composition, but also take into account the specific method of the actually committed encroachment. It is proposed to clarify in the legislative definition of an accomplice that a person who contributed to the commission of a crime by eliminating obstacles is recognized as an accomplice only if such elimination was carried out at the stage of preparation for the crime.

Doubly constrained totally positive line insertion

It is shown that in any TP matrix, a line (row or column) with two speci˝ed entries in any positions (and the others appropriately chosen) may be inserted in any position, as long as the two entries are consistent with total positivity. This generalizes an unconstrained result previously proven, and the two may not generally be increased to three or more. Applications are given, and this fact should be useful in other completion problems, as the unconstrained result has been.