Weak duality

Abstract: The optimal power flow (OPF) problem is nonconvex and generally hard to solve. In this paper, we propose a semidefinite programming (SDP) optimization, which is the dual of an equivalent form of the OPF problem. A global optimum solution to the OPF problem can be retrieved from a solution of this convex dual problem whenever the duality gap is zero. A necessary and sufficient condition is provided in this paper to guarantee the existence of no duality gap for the OPF problem. This condition is satisfied by the standard IEEE benchmark systems with 14, 30, 57, 118, and 300 buses as well as several randomly generated systems. Since this condition is hard to study, a sufficient zero-duality-gap condition is also derived. This sufficient condition holds for IEEE systems after small resistance (10-5 per unit) is added to every transformer that originally assumes zero resistance. We investigate this sufficient condition and justify that it holds widely in practice. The main underlying reason for the successful convexification of the OPF problem can be traced back to the modeling of transformers and transmission lines as well as the non-negativity of physical quantities such as resistance and inductance.

Abstract: Consider a communication system whereby multiple users share a common frequency band and must choose their transmit power spectral densities dynamically in response to physical channel conditions. Due to co-channel interference, the achievable data rate of each user depends on not only the power spectral density of its own, but also those of others in the system. Given any channel condition and assuming Gaussian signaling, we consider the problem to jointly determine all users' power spectral densities so as to maximize a system-wide utility function (e.g., weighted sum-rate of all users), subject to individual power constraints. For the discretized version of this nonconvex problem, we characterize its computational complexity by establishing the NP-hardness under various practical settings, and identify subclasses of the problem that are solvable in polynomial time. Moreover, we consider the Lagrangian dual relaxation of this nonconvex problem. Using the Lyapunov theorem in functional analysis, we rigorously prove a result first discovered by Yu and Lui (2006) that there is a zero duality gap for the continuous (Lebesgue integral) formulation. Moreover, we show that the duality gap for the discrete formulation vanishes asymptotically as the size of discretization decreases to zero.

Abstract: A theory of “discrete convex analysis” is developed for integer-valued functions defined on integer lattice points. The theory parallels the ordinary convex analysis, covering discrete analogues of the fundamental concepts such as conjugacy, subgradients, the Fenchel min-max duality, separation theorems and the Lagrange duality framework for convex/nonconvex optimization. The technical development is based on matroid-theoretic concepts, in particular, submodular functions and exchange axioms. Sections 1–4 extend the conjugacy relationship between submodularity and exchange ability, deepening our understanding of the relationship between convexity and submodularity investigated in the eighties by A. Frank, S. Fujishige, L. Lovasz and others. Sections 5 and 6 establish duality theorems for M- and L-convex functions, namely, the Fenchel min-max duality and separation theorems. These are the generalizations of the discrete separation theorem for submodular functions due to A. Frank and the optimality criteria for the submodular flow problem due to M. Iri-N. Tomizawa, S. Fujishige, and A. Frank. A novel Lagrange duality framework is also developed in integer programming. We follow Rockafellar’s conjugate duality approach to convex/nonconvex programs in nonlinear optimization, while technically relying on the fundamental theorems of matroid-theoretic nature.

Abstract: A general quantum field theory is considered in which the fields are assumed to be operator‐valued tempered distributions The system of fields may include any number of boson fields and fermion fields A theorem which relates certain complex Lorentz transformations to the T C P transformation is stated and proved With reference to this theorem, duality conditions are considered, and it is shown that such conditions hold under various physically reasonable assumptions about the fields Extensions of the algebras of field operators are discussed with reference to the duality conditions Local internal symmetries are discussed, and it is shown that these commute with the Poincare group and with the T C P transformation