Jerome Bracken

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.

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.

Lanchester models of the Ardennes campaign

A detailed data base of the Ardennes campaign of World War II (December 15, 1944 through January 16, 1945) has recently been developed. The present article formulates four Lanchester models of the campaign and estimates their parameters for these data. Two-sided time histories of warfare on battles and campaigns are very rare, so Lanchester models have seldom been validated with historical data. The models are homogeneous in that tanks, armored personnel carriers, artillery, and manpower are weighted to yield a measure of strength of the Allied and German forces. This weighting is utilized for combat power and for losses. The models treat combat forces in the campaign (including infantry, armor, and artillery manpower) and total forces in the campaign (including both combat manpower and support manpower.) Four models are presented. Two models have five parameters (Allied individual effectiveness, German individual effectiveness, exponent of shooting force, exponent of target force, and a tactical parameter reflecting which side is defending and attacking.) The other two models remove the tactical parameter, which is not generally known prior to warfare, and estimate the other parameters without the tactical parameter. The main results of the research are (a) the Lanchester linear model fits the Ardennes campaign data in all four cases, and (b) when combat forces are considered Allied individual effectiveness is greater than German individual effectiveness, whereas when total forces are considered Allied and German individual effectiveness is the same. The interpretation of the latter result is that the two sides had essentially the same individual capabilities but were organized differently—the Allies chose to have more manpower in the support forces, which yielded greater individual capabilities in the combat forces. The overall superiority of the Allies in the campaign led to the attrition to the Allies being a smaller portion of their forces.

A Strategic Weapons Exchange Allocation Model

We formulate a two-strike strategic weapons exchange, model as a max-min problem with the first strikers allocation affecting the second strikers feasible region. The max-min problem is shown to be equivalent to a separable, nonconvex program, to which an algorithm designed to locate an approximate global solution is then applied. The solution of three example problems is given.

Robust preallocated preferential defense

Abstract : The problem is to protect a set of T identical targets that may come under attack by A identical weapons. The targets are to be defended by D identical interceptors, which must be preallocated to defend selected targets. The attacker is aware of the number of interceptors, but is ignorant of their allocation. The size of the attack is chosen by the attacker from within a specified range. The robust strategies developed in this paper do not require the defender to assume an attack size. Rather, the defender chooses a strategy which is good over a wide range of attack sizes, though not necessarily best for any particular attack size. The attacker, knowing that the defender is adopting a robust strategy, chooses the optimal attack strategy for the number of weapons he chooses to expend. The expected number of survivors is a function of the robust defense strategy and optimal attack strategy against this robust defense.

Technical Note-A Method for Solving Mathematical Programs with Nonlinear Programs in the Constraints

This note describes a computational procedure for solving mathematical programs with nonlinear programs in the constraints; it uses the SUMT computer program for the overall mathematical program, and a new program called INSUMT for the nonlinear programs in the constraints. IN AN EARLIER paper,R1] we formulated a class of mathematical programs with optimization problems in the constraints, presented theory dealing with properties of the relevant functions that result in convex programming problems and discussed interpretations. It gives an example with linear programs in the constraints.

Robustness of Preallocated Preferential Defense with Assumed Attack Size and Perfect Attacking and Defending Weapons

The problem is to protect a set of t targets by n perfect interceptors against an attack by m perfect weapons. If the defender solves for an optimal preallocated preferential defense and associated game value assuming m1 attackers, and the attacker knows the assumption of the defender and utilized m2 attackers, he may be able to achieve significantly more damage than the defender assumed that there would be m2 attackers. The paper treats the robustness of preallocated preferential defense to assumptions about the size of the attack and presents results of an alternative approach.

Worldwide nuclear coalition games: a valuation of strategic offensive and defensive forces

Methods are proposed for the valuation of strategic offensive and defensive force structures, with emphasis on the consideration of incentives for the formation of coalitions. Coalitions consist of subsets of the nuclear weapons states, together with the nonnuclear weapons states taken as components of the total value target inventory. The basic approach is to formulate and solve two worldwide nuclear coalition games. In the first game, the first striking coalition is retaliated against by the surviving weapons of the second striking coalition, minimizing the objective function of the first striking coalition. In the second game, the surviving weapons of the second striking coalition are used to maximize its own objective function. The objective function in both models is the percent of surviving value. The games differ substantially. Computational results are presented for all possible coalitions of nuclear weapons states and neutrals. Offensive weapons, defensive weapons, and value target data bases are...

Two Models for Optimal Allocation of Aircraft Sorties

This paper presents two models for allocating general-purpose aircraft to missions in a multiperiod war. The models are two-person, zero-sum, sequential games with simultaneous moves each period. Ground forces as well as air forces are included. Three measures of effectiveness are available. The paper treats both a game allowing nonadaptive strategies and a game allowing behavioral strategies. It is shown that the latter game is equivalent to a game allowing the larger class of adaptive strategies.

Technical Note-The Equivalence of Two Mathematical Programs with Optimization Problems in the Constraints

Abstract : Two classes of mathematical programs with optimization problems in the constraints have recently been studied by two of the authors. The first class involves mathematical programs in the constraints, and the second class involves max-min problems in the constraints. A computational technique has been developed and shown to be effective in solving problems of the first class. The authors show that the computational technique can be applied to problems of the apparently wider second class.