Minimax model of transport operations of emergency on-orbit servicing in heliosynchronous orbits

Heliosynchronous orbits are attractive for space system construction. As a result, the number of spacecraft operating therein is constantly increasing. To increase their efficiency, timely on-orbit servicing (both scheduled and emergency) is needed. Emergency on-orbit servicing of spacecraft is needed in the case of unforeseen, emergency situations with them. According to available statistical estimates, emergency situations with serviced spacecraft are not frequent. Because of this, serviced spacecraft must be within the reach of a service spacecraft for a long time. In planning emergency on-orbit servicing, the following limitations must be met: the time it takes the service spacecraft to approach any of the serviced spacecraft must not exceed its allowable value, and the service spacecraft’s allowable energy consumption must not be exceeded. This paper addresses the problem of searching for emergency on-orbit servicing that would be allowable in terms of time and energy limitations and would meet technical and economical constraints. The aim of this work is to develop a mathematical constrained optimization model for phasing orbit parameter choice, whose use would allow one to minimize the maximum time of transport operations in emergency on-orbit servicing of a spacecraft group in the region of heliosynchronous orbits. The problem is solved by constrained minimax optimization. What is new is the formulation of a minimax (guaranteeing) criterion for choosing phasing orbit parameters that minimize the maximum time of emergency on-orbit servicing transport operations. In the minimax approach, the problem is formulated as the problem of searching for the best solution such that the result is certain to be attained for any allowable sets of indeterminate factors. The proposed mathematical model may be used in planning emergency on-orbit service operations to minimize the maximum duration of emergency on-orbit servicing transport operations due to a special choice of the service spacecraft phasing and parking orbit parameters.

Box-constrained optimization for minimax supervised learning

In this paper, we present the optimization procedure for computing the discrete boxconstrained minimax classifier introduced in [1, 2]. Our approach processes discrete or beforehand discretized features. A box-constrained region defines some bounds for each class proportion independently. The box-constrained minimax classifier is obtained from the computation of the least favorable prior which maximizes the minimum empirical risk of error over the box-constrained region. After studying the discrete empirical Bayes risk over the probabilistic simplex, we consider a projected subgradient algorithm which computes the prior maximizing this concave multivariate piecewise affine function over a polyhedral domain. The convergence of our algorithm is established.

Algebraic solution of a problem of optimal project scheduling in project management

A problem of optimal scheduling is considered for a project that consists of a certain set of works to be performed under given constraints on the times of start and finish of the works. As the optimality criterion for scheduling, the maximum deviation of the start time of works is taken to be minimized. Such problems arise in project management when it is required, according to technological, organizational, economic or other reasons, to provide, wherever possible, simultaneous start of all works. The scheduling problem under consideration is formulated as a constrained minimax optimization problem and then solved using methods of tropical (idempotent) mathematics which deals with the theory and applications of semirings with idempotent addition. First, a tropical optimization problem is investigated defined in terms of a general idempotent semifield (an idempotent semiring with invertible multiplication), and a complete analytical solution of the problem is derived. The result obtained is then applied to find a direct solution of the scheduling problem in a compact vector form ready for further analysis of solutions and straightforward computations. As an illustration, a numerical example of solving optimal scheduling problem is given for a project that consists of four works.

An Implementable SAA Nonlinear Lagrange Algorithm for Constrained Minimax Stochastic Optimization Problems

This paper proposes an implementable SAA (sample average approximation) nonlinear Lagrange algorithm for the constrained minimax stochastic optimization problem based on the sample average approximation method. A computable nonlinear Lagrange function with sample average approximation functions of original functions is minimized and the Lagrange multiplier is updated based on the sample average approximation functions of original functions in the algorithm. And it is shown that the solution sequences obtained by the novel algorithm for solving subproblem converge to their true counterparts with probability one as the sample size approximates infinity under some moderate assumptions. Finally, numerical experiments are carried out for solving some typical test problems and the obtained numerical results preliminarily demonstrate that the proposed algorithm is promising.