This paper studies the representation of a non-negative polynomial f on a non-compact semi-algebraic set K modulo its KKT (Karush-KuhnTucker) ideal. Under the assumption that f satisfies the boundary Hessian conditions (BHC) at each zero of f in K; we show that f can be represented as a sum of squares (SOS) of real polynomials modulo its KKT ideal if f ≥ 0 on K.

This paper presents a voltage stability constrained optimal power flow that is expressed via a bilevel programming framework. The inner objective function is dedicated for maximizing voltage stability margin while the outer objective function is focused on minimization of total production cost of thermal units. The original two stage problem is converted to a single level optimization problem v...

Robust model predictive control approaches and other applications lead to nonlinear optimization problems defined on (scenario) trees. We present structure-preserving Quasi-Newton update formulas as well as structured inertia correction techniques that allow to solve these problems by interior-point methods with specialized KKT solvers for tree-structured optimization problems. The same type of...

Convex optimization solvers for embedded systems find widespread use. This letter presents a novel technique to reduce the run-time of decomposition of KKT matrix for the convex optimization solver for an embedded system, by two orders of magnitude. We use the property that although the KKT matrix changes, some of its block sub-matrices are fixed during the solution iterations and the associate...

Using the fact that any two player discounted stochastic game with finite state and action spaces can be recast as a non-convex constrained optimization problem, where each global minima corresponds to a stationary Nash equilibrium, we present a sequential quadratic programming based algorithm that converges to a KKT point. This KKT point is an -Nash equilibrium for some > 0 and under some suit...

In barrier methods for constrained optimization, the main work lies in solving large linear systems Kp = r, where K is symmetric and indefinite. For linear programs, these KKT systems are usually reduced to smaller positive-definite systems AHAq = s, where H is a large principal submatrix of K. These systems can be solved more efficiently, but AHA is typically more ill-conditioned than K. In or...

The Lagrange dual function is: g(u, v) = min x L(x, u, v) The corresponding dual problem is: maxu,v g(u, v) subject to u ≥ 0 The Lagrange dual function can be viewd as a pointwise maximization of some affine functions so it is always concave. The dual problem is always convex even if the primal problem is not convex. For any primal problem and dual problem, the weak duality always holds: f∗ ≥ g...