No AccessEngineering NotesMars Entry Trajectory Planning with Range Discretization and Successive ConvexificationXu Liu, Shuang Li Ming XinXu LiuNanjing University of Aeronautics Astronautics, 211106 Nanjing, People’s Republic China*Ph.D. Candidate, Department Aerospace Control Engineering; .Search for more papers by this author, https://orcid.org/0000-0001-9142-5036Nanjing China†Professor, (Co...

This paper presents a new method of training neural networks including deep learning machines, which is based on the idea of convexifying the training error criterion by the use of the risk-averting error (RAE) criterion. Convexification creates tunnels between the depressed regions around saddle points, tilts the plateaus, and eliminates nonglobal local minima. The difficulties in computing th...

Let us define for a compact set A ⊂ R the sequence A(k) = { a1 + · · ·+ ak k : a1, . . . , ak ∈ A } = 1 k ( A+ · · ·+A } {{ } k times ) . It was independently proved by Shapley, Folkman and Starr (1969) and by Emerson and Greenleaf (1969) that A(k) approaches the convex hull of A in the Hausdorff distance induced by the Euclidean norm as k goes to ∞. We explore in this survey how exactly A(k) a...

Given a polygon P in the plane, a pop operation is the reflection of a vertex with respect to the line through its adjacent vertices. We define a family of alternating polygons, and show that any polygon from this family cannot be convexified by pop operations. This family contains simple, as well as non-simple (i.e., self-intersecting) polygons, as desired. We thereby answer in the negative an...

Optimal control problems are common in aerospace engineering. A Python software program called PySCP is described for solving multiple-phase optimal using sequential convex programming methods. By constructing a series of approximated second-order cone subproblems, approaches to the solution original problem an iterative way. The key components detail, including convexification, discretization,...

We study a continuous version of the capacity and flow assignment problem (CFA) where the design cost is combined with an average delay measure to yield a non convex objective function coupled with multicommodity flow constraints. A separable convexification of each arc cost function is proposed to obtain approximate feasible solutions within easily computable gaps from optimality. On the other...

Every function of several variables with the continuous second derivative can be convexified (i.e., made convex) by adding to it a quadratic “convexifier”. In this paper we give simple estimates on the bounds of convexifiers. Using the idea of convexification, many problems in applied mathematics can be reduced to convex mathematical programming models. This is illustrated here for nonlinear pr...

For a class of infinite-dimensional minimization problems with nonlinear equality constraints, an iterative algorithm for finding global solutions is suggested. A key assumption is the convexity of the “epigraph”, a set in the product of the image spaces of the constraint and objective functions. A convexification method involving randomization is used. The algorithm is based on the extremal sh...

The global optimisation of non-convex quadratically constrained quadratic programs is a notoriously difficult problem, being not only NP-hard in the strong sense, but also very difficult in practice. We present a new heuristic approach to this problem, which enables one to obtain solutions of good quality in reasonable computing times. The heuristic consists of four phases: binarisation, convex...

Abstract Solving mixed-integer nonlinear optimization problems (MINLPs) to global optimality is extremely challenging. An important step for enabling their solution consists in the design of convex relaxations feasible set. Known approaches based on spatial branch-and-bound become more effective tighter used are. Relaxations are commonly established by underestimators, where each constraint fun...