On the Optimal Strategic Defense Against Invasion using Acquired Intelligence Wei Chen, Dimitrios Dimitriadis, Debdipta Goswami and Nirupam Gupta Abstract—In this project, we have explored a potential utility of convex optimization in designing optimal strategic defense against an invading enemy using the acquired intelligence on the attacks. This situation often arises when one of the belligerent entity has sufficient intelligence about the opponent’s move, but does not want to rise suspicion in order to retain the tactical advantage. We have developed the problem as a cost-minimization robust programing, which involves second-order cone (SOC) constraints and additional boolean (non-convex) constraints. The problem is solved separately with two different convex relaxations and the results are compared with the branch and bound method, which is widely used for mixed integer quadratically constrained programming (MIQCP) problems. I. I NTRODUCTION From the early dawn of warfare, war strategists mostly depended on their well-earned intuitions to make an effec- tive defense plan. With advances in mathematical horizons and computational capabilities, it become has possible for war strategists to acid test their plan before execution up to certain extent. Our proposed problem is inspired by the 2014 movie ‘The Imitation Game’ be Morten Tyldum, in which Allied forces gained tactical advantage over the Axis forces by breaking the Enigma code. Enigma code was widely used by Axis’s for transmitting their instructions, which often contained geographical locations and timings of their eventual attacks. To hide this advantage from their enemies the allied forces decided to use this tactical advantage cleverly and instead of stopping every single attack, only certain attacks were neutralized. Defending each attack launched by the Axis powers raise suspicion in the Axis high-command, whereas not defending an attack increases damage (cost) to the Allies. So the objective of the Allies is to minimize the cost while keeping the suspicion well below the threshold. II. NOTATIONS Throughout the paper, the sets R, R n , S n and R n×n denote the sets of real-valued scalars, n-dimensional real-valued vectors, symmetric and n by n matrices with real-valued elements respectively. The notation ‖·‖ is used to denote both vector 2-norm and induced 2-norm of a matrix. For real-valued square matrices of appropriate dimensions A and B, A  B means that the matrix A − B is positive semi-definite. tr(A) represents the trace of a square matrix A. III. PROBLEM FORMULATION Here the two belligerent entities Allied ForcesA and Axis power G are competing with conflicting interest. Due to prior intelligence (breaking of Enigma) A has the upper hand in knowing G’s strategy and wants to minimise the cost (damage in terms of causalty or property loss) by defending the attacks. But as each attack by G is neutralised, G gets suspicious about the invulnerability of their Enigma code, and hence A does not want to raise suspicion at an alarming level so that G understands their code is compromised and A loses its advantage. A. Assumptions The problem is formulated with following assumptions: • There are two belligerent parties: The Allied (A) and the Axis (G). • Aggregate cost c i associated with attack i by G is known to A for all i =1,...,n. • G associates suspicion cost with each attack which de- pends on the possibility of the attack getting neutralized to check security of their chosen enigma setting/code. Ex- act suspicion cost associated with each attack is unknown to A, but A has some knowledge of these costs and uses a biased uncertain suspicion cost vector α ∈ R n , given as α =˜ α + Mu where ¯ α, M ∈ R n×n are known to A and u ∈ R n is a bounded uncertainty with ‖u‖≤ 1. • Let x ∈ R n be the decision vector, x i = −1 implies, attack i should be neutralized and vice-versa. To evade suspicion of successful cryptanalysis of the enigma set- ting, A makes sure that j  i=1 α i (−x i ) ≤ s, j =1,...,n, ∀α ∈ E where, E = { ˜ α + Mu : ‖u‖≤ 1}. This ensures that the cumulative suspicion always lies under the threshold. • A has limited resources r, and defending i-th attack demands p i amount of resources. Hence p T (1 − x) ≤ r • In the beginning the suspicion threshold s is assumed constant and known to A. The initial optimization problem will be a robust LP prob- lem along with binary variable x. minimize x∈R n 1 2 c T (x + 1) subject to j  i=1 α i (−x i ) ≤ s, j =1,...,n, ∀α ∈ E x 2 i =1,i =1,...,n. p T (1 − x) ≤ r (1)