We introduce a vertical type relaxation for optimal control problems which only have $L^1$-coercivity for their controls. Usually such problems feature both concentration and oscillation effects at the same time. We propose relaxing to an associated problem in space-time, where the controls can be considered bounded in $L^\infty$, greatly simplifying any analysis. In this relaxation, concentrations are transformed into vertical parts and oscillations can be dealt with using Young-measures. This technique can be extended to similar problems on infinite-dimensional spaces.
We obtain a measure representation for a functional arising in the context of optimal design problems under linear growth conditions. The functional in question corresponds to the relaxation with respect to a pair $(χ,u)$, where $χ$ is the characteristic function of a set of finite perimeter and $u$ is a function of bounded deformation, of an energy with a bulk term depending on the symmetrised gradient as well as a perimeter term.
We consider a nonlinear optimal control problem with dynamics described by a differential inclusion involving a maximal monotone map $A:\mathbb{R}^N\rightarrow2^{\mathbb{R}^N}$. We do not assume that $D(A)=\mathbb{R}^N$, incorporating in this way systems with unilateral constraints in our framework. We present two relaxation methods. The first one is an outgrowth of the reduction method from the existence theory, while the second method uses Young measures. We show that the two relaxation methods are equivalent and admissible.
Computational speed and global optimality are key needs for practical algorithms for the optimal power flow problem. Two convex relaxations offer a favorable trade-off between the standard second-order cone and the standard semidefinite relaxations for large-scale meshed networks in terms of optimality gap and computation time: the tight-and-cheap relaxation (TCR) and the quadratic convex relaxation (QCR). We compare these relaxations on 60 PGLib-OPF test cases with up to 1,354 buses under three operating conditions and show that TCR dominates QCR on all 20 typical (TYP) test cases, on 18 out …
The relaxation systems are an important subclass of the passive systems that arise naturally in applications. We exploit the fact that they have highly structured state-space realisations to derive analytical solutions to some simple H-infinity type optimal control problems. The resulting controllers are also relaxation systems, and often sparse. This makes them ideal candidates for applications in large-scale problems, which we demonstrate by designing simple, sparse, electrical circuits to optimally control large inductive networks and to solve linear regression problems.
The optimal reactive power dispatch (ORPD) problem is an alternating current optimal power flow (ACOPF) problem where discrete control devices for regulating the reactive power, such as shunt elements and tap changers, are considered. The ORPD problem is modelled as a mixed-integer nonlinear optimization problem and its complexity is increased compared to the ACOPF problem, which is highly nonconvex and generally hard to solve. Recently, convex relaxations of the ACOPF problem have attracted a significant interest since they can lead to global optimality. We propose a tight conic relaxation of…
This paper proves that in an unbalanced multi-phase network with a tree topology, the semidefinite programming relaxation of optimal power flow problems is exact when critical buses are not adjacent to each other. Here a critical bus either contributes directly to the cost function or is where an injection constraint is tight at optimality. Our result generalizes a sufficient condition for exact relaxation in single-phase tree networks to tree networks with arbitrary number of phases.
The classical alternating current optimal power flow problem is highly nonconvex and generally hard to solve. Convex relaxations, in particular semidefinite, second-order cone, convex quadratic, and linear relaxations, have recently attracted significant interest. The semidefinite relaxation is the strongest among them and is exact for many cases. However, the computational efficiency for solving large-scale semidefinite optimization is lower than for second-order cone optimization. We propose a conic relaxation obtained by combining semidefinite optimization with the reformulation-linearizati…
Semidefinite relaxation techniques have shown great promise for nonconvex optimal power flow problems. However, a number of independent numerical experiments have led to concerns about scalability and robustness of existing SDP solvers. To address these concerns, we investigate some numerical aspects of the problem and compare different state-of-the-art solvers. Our results demonstrate that semidefinite relaxations of large problem instances with on the order of 10,000 buses can be solved reliably and to reasonable accuracy within minutes. Furthermore, the semidefinite relaxation of a test cas…
This paper proposes a general fixture layout design framework that directly integrates the system equation with the convex relaxation method. Note that the optimal fixture design problem is a large-scale combinatorial optimization problem, we relax it to a convex semidefinite programming (SDP) problem by adopting sparse learning and SDP relaxation techniques. It can be solved efficiently by existing convex optimization algorithms and thus generates a near-optimal fixture layout. A real case study in the half-to-half fuselage assembly process indicates the superiority of our proposed algorithm …
An optimal transport problem on finite spaces is a linear program. Recently, a relaxation of the optimal transport problem via strictly convex functions, especially via the Kullback--Leibler divergence, sheds new light on data sciences. This paper provides the mathematical foundations and an iterative process based on a gradient descent for the relaxed optimal transport problem via Bregman divergences.
Optimal power flow (OPF) is one of the key electric power system optimization problems. "Moment" relaxations from the Lasserre hierarchy for polynomial optimization globally solve many OPF problems. Previous work illustrates the ability of higher-order moment relaxations to approach the convex hulls of OPF problems' non-convex feasible spaces. Using a small test case, this paper focuses on the ability of the moment relaxations to globally solve problems with objective functions that have unconstrained minima at infeasible points inside the convex hull of the non-convex constraints.
This article presents a new method for computing guaranteed convex and concave relaxations of nonlinear stochastic optimal control problems with final-time expected-value cost functions. This method is motivated by similar methods for deterministic optimal control problems, which have been successfully applied within spatial branch-and-bound (B&B) techniques to obtain guaranteed global optima. Relative to those methods, a key challenge here is that the expected-value cost function cannot be expressed analytically in closed form. Nonetheless, the presented relaxations provide rigorous lower and…
The first part of this paper proposed a family of penalized convex relaxations for solving optimization problems with bilinear matrix inequality (BMI) constraints. In this part, we generalize our approach to a sequential scheme which starts from an arbitrary initial point (feasible or infeasible) and solves a sequence of penalized convex relaxations in order to find feasible and near-optimal solutions for BMI optimization problems. We evaluate the performance of the proposed method on the H2 and Hinfinity optimal controller design problems with both centralized and decentralized structures. Th…
The alternating-current optimal power flow (ACOPF) is one of the best known non-convex non-linear optimisation problems. We present a novel re-formulation of ACOPF, which is based on lifting the rectangular power-voltage rank-constrained formulation, and makes it possible to derive alternative SDP relaxations. For those, we develop a first-order method based on the parallel coordinate descent with a novel closed-form step based on roots of cubic polynomials.
Nonconvex optimal-control problems governed by evolution problems in infinite-dimensional spaces (as e.g. parabolic boundary-value problems) needs a continuous (and possibly also smooth) extension on some (preferably convex) compactification, called relaxation, to guarantee existence of their solutions and to facilitate analysis by relatively conventional tools. When the control is valued in some subsets of Lebesgue spaces, the usual extensions are either too coarse (allowing in fact only very restricted nonlinearities) or too fine (being nonmetrizable). To overcome these drawbacks, a compromi…
This paper addresses the problem of solving a class of nonlinear optimal control problems (OCP) with infinite-dimensional linear state constraints involving Riesz-spectral operators. Each instance within this class has time/control dependent polynomial Lagrangian cost and control constraints described by polynomials. We first perform a state-mode discretization of the Riesz-spectral operator. Then, we approximate the resulting finite-dimensional OCPs by using a previously known hierarchy of semidefinite relaxations. Under certain compactness assumptions, we provide a converging hierarchy of se…
This paper is concerned with optimal power flow (OPF), which is the problem of optimizing the transmission of electricity in power systems. Our main contributions are as follows: (i) we propose a novel parabolic relaxation, which transforms non-convex OPF problems into convex quadratically-constrained quadratic programs (QCQPs) and can serve as an alternative to the common practice semidefinite programming (SDP) and second-order cone programming (SOCP) relaxations, (ii) we propose a penalization technique which is compatible with the SDP, SOCP, and parabolic relaxations and guarantees the reco…
In this paper, we investigate optimal control problems governed by semilinear elliptic variational inequalities involving constraints on the state, and more precisely the obstacle problem. Since we adopt a numerical point of view, we first relax the feasible domain of the problem, then using both mathematical programming methods and penalization methods we get optimality conditions with smooth Lagrange multipliers. Some numerical experiments using IPOPT algorithm are presented to verify the efficiency of our approach.
This paper develops a novel second order cone relaxation of the semidefinite programming formulation of optimal power flow, that does not imply the `angle relaxation'. We build on a technique developed by Kim et al., extend it for complex matrices, and apply it to 3x3 positive semidefinite matrices to generate novel second-order cone constraints that augment upon the well-known 2x2 principal-minor based second-order cone constraints. Finally, we apply it to optimal power flow in meshed networks and provide numerical illustrations.
Optimal power flow (OPF) is an important problem in the operation of electric power systems. Due to the OPF problem's non-convexity, there may exist multiple local optima. Certifiably obtaining the global solution is important for certain applications of OPF problems. Many global optimization techniques compute an optimality gap that compares the achievable objective value corresponding to the feasible point from a local solution algorithm with the objective value bound from a convex relaxation technique. Rather than the traditional practice of completely separating the local solution and conv…
In this paper, we consider the optimal power flow (OPF) problem which consists in determining the power production at each bus of an electric network by minimizing the production cost. Our contribution is an exact solution algorithm for the OPF problem. It consits in a spatial branch-and-bound algorithm based on a compact quadratically constrained convex relaxation. This compact relaxation is computed by solving the rank relaxation once at the beginning of the algorithm. The key point of this approach is that the lower bound at the root node of the branch-and-bound tree is equal to the rank re…
This paper proposes a set of closed-form conditions to ensure the strong duality of second-order cone program (SOCP) formulation for AC power flow in radial power networks. In addition, numerical evaluations on IEEE 33-bus test networks and a real-world distribution network are performed to demonstrate the validity of the proposed conditions.
In this letter, we propose a data-driven warm start approach, empowered by artificial neural networks, to boost the efficiency of convex relaxations in optimal gas flow. Case studies show that this approach significantly decreases the number of iterations for the convex-concave procedure algorithm, and optimality and feasibility of the solution can still be guaranteed.
We study the equivalence of several well-known sufficient optimality conditions for a general quadratically constrained quadratic program (QCQP). The conditions are classified in two categories. The first one is for determining an optimal solution and the second one is for finding an optimal value. The first category of conditions includes the existence of a saddle point of the Lagrangian function and the existence of a rank-1 optimal solution of the primal SDP relaxation of QCQP. The second category includes $η_p = ζ$, $η_d = ζ$, and $φ= ζ$, where $ζ$, $η_p$, $η_d$, and $φ$ denote the optimal…
This paper considers the optimal control for hybrid systems whose trajectories transition between distinct subsystems when state-dependent constraints are satisfied. Though this class of systems is useful while modeling a variety of physical systems undergoing contact, the construction of a numerical method for their optimal control has proven challenging due to the combinatorial nature of the state-dependent switching and the potential discontinuities that arise during switches. This paper constructs a convex relaxation-based approach to solve this optimal control problem. Our approach begins…
