Immunizing Conic Quadratic Optimization Problems Against Implementation Errors

Abstract

We show that the robust counterpart of a convex quadratic constraint with ellipsoidal implementation error is equivalent to a system of conic quadratic constraints. To prove this result we first derive a sharper result for the S-lemma in case the two matrices involved can be simultaneously diagonalized. This extension of the S-lemma may also be useful for other purposes. We extend the result to the case in which the uncertainty region is the intersection of two convex quadratic inequalities. The robust counterpart for this case is also equivalent to a system of conic quadratic constraints. Results for convex conic quadratic constraints with implementation error are also given. We conclude with showing how the theory developed can be applied in robust linear optimization with jointly uncertain parameters and implementation errors, in sequential robust quadratic programming, in Taguchi's robust approach, and in the adjustable robust counterpart.

Robust Optimization with Ambiguous Stochastic Constraints Under Mean and Dispersion Information

“…For (26) to be tractable, the functions g i (x + z) need to be affine in z or need to belong to one of the special cases considered in Ben-Tal and den Hertog (2011). Similarly, one can reformulate a problem where multiplicative error occurs, i.e., where…”

Hidden Convexity in Partially Separable Optimization

“…. , f n , and the structure of the constraints in (3). In a first step we group some general facts about problem (3) and its relaxation (5).…”

“…As was observed in [3], when the matrices B, D l (l ∈ L) can be simultaneously diagonalized, problem (40) can be cast as an instance of problem (3), where we choose the functions f i (x i ) = x 2 i . Indeed, if U T BU, U T D l U are all diagonal, then we can make a linear change of variables (replacing x by U −1 x), and reformulate (40) as…”

“…Moreover, if, in the quadratic optimization problem (40), the matrices B, D l (l ∈ l) are all linear combinations of two matrices, one of them definite, then the problem can be reformulated as an instance of (41). This is the case, e.g., when Q B ∩ Q D 1 = {0} and we are in one of the following cases: The following example, taken from [3], illustrates the above case (ii) where one of the two constraints is linear. Namely, if an optimization problem has a quadratic objective function and a conic quadratic constraint of the form:…”

“…Hidden convexity was discovered for problems with indefinite quadratic objective functions, and one or two quadratic constraints( [10], [13]). The hidden convex problems in these cases are generally Semi Definite Programs (SDP), or, in more specific cases, are Conic Quadratic Programs (CQP) ( [3]). The main mathematical tools to derive these results are basic results on the range space of (indefinite) quadratic forms ( [11]) and the so-called S-lemma ( [12], [8]).…”