Nonmonotone curvilinear line search methods for unconstrained optimization
Computational Optimization and Applications
Convergence to Second Order Stationary Points in Inequality Constrained Optimization
Mathematics of Operations Research
A Complementary Pivoting Approach to the Maximum Weight Clique Problem
SIAM Journal on Optimization
On Some Properties of Quadratic Programs with a Convex Quadratic Constraint
SIAM Journal on Optimization
SIAM Journal on Optimization
Evolution towards the Maximum Clique
Journal of Global Optimization
On Standard Quadratic Optimization Problems
Journal of Global Optimization
A clique algorithm for standard quadratic programming
Discrete Applied Mathematics
A Variational Approach to Copositive Matrices
SIAM Review
Journal of Global Optimization
Standard bi-quadratic optimization problems and unconstrained polynomial reformulations
Journal of Global Optimization
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A standard quadratic optimization problem (StQP) consists of finding the largest or smallest value of a (possibly indefinite) quadratic form over the standard simplex which is the intersection of a hyperplane with the positive orthant. This NP-hard problem has several immediate real-world applications like the Maximum-Clique Problem, and it also occurs in a natural way as a subproblem in quadratic programming with linear constraints. To get rid of the (sign) constraints, we propose a quartic reformulation of StQPs, which is a special case (degree four) of a homogeneous program over the unit sphere. It turns out that while KKT points are not exactly corresponding to each other, there is a one-to-one correspondence between feasible points of the StQP satisfying second-order necessary optimality conditions, to the counterparts in the quartic homogeneous formulation. We supplement this study by showing how exact penalty approaches can be used for finding local solutions satisfying second-order necessary optimality conditions to the quartic problem: we show that the level sets of the penalty function are bounded for a finite value of the penalty parameter which can be fixed in advance, thus establishing exact equivalence of the constrained quartic problem with the unconstrained penalized version.