Two-sided Optimization Research Articles

One-Sided Optimal Assignment and Swap Algorithm for Two-Sided Optimization of Assignment Problem

Generally, the optimal solution of assignment problem can be obtained by Hungarian algorithm of two-sided optimization with time complexity . This paper suggests one-sided optimal assignment and swap optimization algorithm with time complexity can be achieve the goal of two-sided optimization. This algorithm selects the minimum cost for each row, and reassigns over-assigned to under-assigned cell. Next, that verifies the existence of swap optimization candidates, and swap optimizes with . For 27 experimental data, the swap-optimization performs only 22% of data, and 78% of data can be get the two-sided optimal result through one-sided optimal result. Also, that can be improves on the solution of best known solution for partial problems.

An Algorithm for Finding an Optimal Projection of a Symmetric Matrix onto a Diagonal Matrix

This work is motivated by two two-sided optimization problems that arise in atomic chemistry when one is looking for a minimal set of localized orbitals with prescribed occupation numbers, respectively, a prescribed total number of electrons. We first propose an optimal analytic solution of the first problem, and then show that an optimal solution of the second problem can be found by solving a convex quadratic programming problem with box constraints and $p$ unknowns. We prove that the latter problem is solved by the active-set method in at most $2p$ iterations. These investigations reveal that the optimal solutions of both problems are projections that are in $\mathcal{C}=\{Y\in\mathbb{R}^{n\times p}:Y^TY=I_p, Y^TN Y=\Delta\}$ for $N$ symmetric and $\Delta$ diagonal and that these solutions are generally nonunique. The question of how one can then select a particular solution out of the set $\mathcal{C}$ leads to the main problem of this paper: the optimization of an arbitrary smooth function whose minimizer describes the solution of interest over $\mathcal{C}$. To solve this problem, we first find that a slight modification of $\mathcal{C}$ is a Riemannian manifold for which geometric objects can be derived that are required to allow optimization over this manifold. Using these geometric tools we propose an augmented Lagrangian-based algorithm that guarantees global convergence to a stationary point of our main problem. The latter is shown by investigating when the LICQ is satisfied. Finally we compare this algorithm numerically with a similar algorithm that, however, does not apply these geometric tools and that is, to our knowledge, not guaranteed to converge. Our results show that our algorithm yields a significantly better performance.

Numerical Solution of Orbital Combat Games Involving Missiles and Spacecraft

This research addresses the problem of the optimal interception of an optimally evasive orbital target by a pursuing spacecraft or missile. The time for interception is to be minimized by the pursuing space vehicle and maximized by the evading target. This problem is modeled as a two-sided optimization problem, i.e. as a two-player zero-sum differential game. The work incorporates a recently developed method, termed “semi-direct collocation with nonlinear programming”, for the numerical solution of dynamic games. The method is based on the formal conversion of the two-sided optimization problem into a single-objective one, by employing the analytical necessary conditions for optimality related to one of the two players. An approximate, first attempt solution for the method is provided through the use of a genetic algorithm in a “preprocessing” phase. Three qualitatively different cases are considered. In the first example the pursuer and the evader are represented by two spacecraft orbiting the Earth in two distinct orbits. The second and the third case involve two missiles, and a missile that pursues an orbiting spacecraft, respectively. The numerical results achieved in this work testify to the robustness and effectiveness of the method also in solving large, complex, three-dimensional problems.

A Two-Sided Optimization for Theater Ballistic Missile Defense

We describe JOINT DEFENDER, a new two-sided optimization model for planning the pre-positioning of defensive missile interceptors to counter an attack threat. In our basic model, a defender pre-positions ballistic missile defense platforms to minimize the worst-case damage an attacker can achieve; we assume that the attacker will be aware of defensive pre-positioning decisions, and that both sides have complete information as to target values, attacking-missile launch sites, weapon system capabilities, etc. Other model variants investigate the value of secrecy by restricting the attacker’s and/or defender’s access to information. For a realistic scenario, we can evaluate a completely transparent exchange in a few minutes on a laptop computer, and can plan near-optimal secret defenses in seconds. JOINT DEFENDER’s mathematical foundation and its computational efficiency complement current missile-defense planning tools that use heuristics or supercomputing. The model can also provide unique insight into the value of secrecy and deception to either side. We demonstrate with two hypothetical North Korean scenarios.

Defense Applications of Mathematical Programs with Optimization Problems in the Constraints

Bracken and McGill have discussed the theory, computations, and an example of mathematical programming models with optimization problems in the constraints [Opns. Res. 21, 37–44 (1973)], and have presented a computer program for solving such models with nonlinear programs in the constraints [Opns. Res. 22, 1097–1101 (1974)]. Bracken, Falk, and McGill have given a procedure for transforming mathematical programs with two-sided optimization problems in the constraints into mathematical programs with nonlinear programs in the constraints [Opns. Res. 22, 1102–1104 (1974)], thus enabling their solution by the computer program. This paper formulates models of defense problems that are convex programs having the mathematical properties treated in the previous papers. The models include several strategic-force-planning models and two general-purpose-force planning models.

Mathematical Programs with Optimization Problems in the Constraints

This paper considers a class of optimization problems characterized by constraints that themselves contain optimization problems. The problems in the constraints can be linear programs, nonlinear programs, or two-sided optimization problems, including certain types of games. The paper presents theory dealing primarily with properties of the relevant functions that result in convex programming problems, and discusses interpretations of this theory. It gives an application with linear programs in the constraints, and discusses computational methods for solving the problems.