Penalized Semidefinite Programming for Quadratically-Constrained Quadratic Optimization

In this paper, we give a new penalized semidefinite programming approach for non-convex quadratically-constrained quadratic programs (QCQPs). We incorporate penalty terms into the objective of convex relaxations in order to retrieve feasible and near-optimal solutions for non-convex QCQPs. We introduce a generalized linear independence constraint qualification (GLICQ) criterion and prove that any GLICQ regular point that is sufficiently close to the feasible set can be used to construct an appropriate penalty term and recover a feasible solution. Inspired by these results, we develop a heuristic sequential procedure that preserves feasibility and aims to improve the objective value at each iteration. Numerical experiments on large-scale system identification problems as well as benchmark instances from the library of quadratic programming (QPLIB) demonstrate the ability of the proposed penalized semidefinite programs in finding near-optimal solutions for non-convex QCQP

Intersection cuts from multiple rows: a disjunctive programming approach

We address the issue of generating cutting planes for mixed integer programs from multiple rows of the simplex tableau with the tools of disjunctive programming. A cut from q rows of the simplex tableau is an intersection cuts from a q-dimensional parametric cross-polytope, which can also be viewed as a disjunctive cut from a 2q-term disjunction. We define the disjunctive hull of the q-row problem, describe its relation to the integer hull, and show how to generate its facets. For the case of binary basic variables, we derive cuts from the stronger disjunctions whose terms are equations. We give cut strengthening procedures using the integrality of the nonbasic variables for both the integer and the binary case. Finally, we discuss some computational experiments.Comment: 38 pages, 6 figure

Active network management for electrical distribution systems: problem formulation, benchmark, and approximate solution

With the increasing share of renewable and distributed generation in electrical distribution systems, Active Network Management (ANM) becomes a valuable option for a distribution system operator to operate his system in a secure and cost-effective way without relying solely on network reinforcement. ANM strategies are short-term policies that control the power injected by generators and/or taken off by loads in order to avoid congestion or voltage issues. Advanced ANM strategies imply that the system operator has to solve large-scale optimal sequential decision-making problems under uncertainty. For example, decisions taken at a given moment constrain the future decisions that can be taken and uncertainty must be explicitly accounted for because neither demand nor generation can be accurately forecasted. We first formulate the ANM problem, which in addition to be sequential and uncertain, has a nonlinear nature stemming from the power flow equations and a discrete nature arising from the activation of power modulation signals. This ANM problem is then cast as a stochastic mixed-integer nonlinear program, as well as second-order cone and linear counterparts, for which we provide quantitative results using state of the art solvers and perform a sensitivity analysis over the size of the system, the amount of available flexibility, and the number of scenarios considered in the deterministic equivalent of the stochastic program. To foster further research on this problem, we make available at http://www.montefiore.ulg.ac.be/~anm/ three test beds based on distribution networks of 5, 33, and 77 buses. These test beds contain a simulator of the distribution system, with stochastic models for the generation and consumption devices, and callbacks to implement and test various ANM strategies

Distributed Mixed-Integer Linear Programming via Cut Generation and Constraint Exchange

Many problems of interest for cyber-physical network systems can be formulated as Mixed-Integer Linear Programs in which the constraints are distributed among the agents. In this paper we propose a distributed algorithmic framework to solve this class of optimization problems in a peer-to-peer network with no coordinator and with limited computation and communication capabilities. At each communication round, agents locally solve a small linear program, generate suitable cutting planes and communicate a fixed number of active constraints. Within the distributed framework, we first propose an algorithm that, under the assumption of integer-valued optimal cost, guarantees finite-time convergence to an optimal solution. Second, we propose an algorithm for general problems that provides a suboptimal solution up to a given tolerance in a finite number of communication rounds. Both algorithms work under asynchronous, directed, unreliable networks. Finally, through numerical computations, we analyze the algorithm scalability in terms of the network size. Moreover, for a multi-agent multi-task assignment problem, we show, consistently with the theory, its robustness to packet loss

Recover Feasible Solutions for SOCP Relaxation of Optimal Power Flow Problems in Mesh Networks

Convex relaxation methods have been studied and used extensively to obtain an optimal solution to the optimal power flow (OPF) problem. Meanwhile, convex relaxed power flow equations are also prerequisites for efficiently solving a wide range of problems in power systems including mixed-integer nonlinear programming (MINLP) and distributed optimization. When the exactness of convex relaxations is not guaranteed, it is important to recover a feasible solution for the convex relaxation methods. This paper presents an alternative convex optimization (ACP) approach that can efficiently recover a feasible solution from the result of second-order cone programming (SOCP) relaxed OPF in mesh networks. The OPF problem is first formulated as a difference-of-convex (DC) programming problem, then efficiently solved by a penalty convex concave procedure (CCP). CCP iteratively linearizes the concave parts of the power flow constraints and solves a convex approximation of the DCP problem. Numerical tests show that the proposed method can find a global or near-global optimal solution to the AC OPF problem, and outperforms those semidefinite programming (SDP) based algorithms.Comment: 8 page

A survey of hidden convex optimization

Motivated by the fact that not all nonconvex optimization problems are difficult to solve, we survey in this paper three widely-used ways to reveal the hidden convex structure for different classes of nonconvex optimization problems. Finally, ten open problems are raised.Comment: 25 page

A note on the split rank of intersection cuts

In this note, we present a simple geometric argument to determine a lower bound on the split rank of intersection cuts. As a first step of this argument, a polyhedral subset of the lattice-free convex set that is used to generate the intersection cut is constructed. We call this subset the restricted lattice-free set. It is then shown that ! log 2(l)mixed integer programming, split rank, intersection cuts.

Multi-Commodity Multi-Facility Network Design

We consider multi-commodity network design models, where capacity can be added to the arcs of the network using multiples of facilities that may have different capacities. This class of mixed-integer optimization models appears frequently in telecommunication network capacity expansion problems, train scheduling with multiple locomotive options, supply chain, and service network design problems. Valid inequalities used as cutting planes in branch-and-bound algorithms have been instrumental in solving large-scale instances. We review the progress that has been done in polyhedral investigations in this area by emphasizing three fundamental techniques. These are the metric inequalities for projecting out continuous flow variables, mixed-integer rounding from appropriate base relaxations and shrinking the network to a small $k$-node graph. The basic inequalities derived from arc-set, cut-set and partition relaxations of the network are also extensively utilized with certain modifications in robust and survivable network design problems

Generation of point sets by convex optimization for interpolation in reproducing kernel Hilbert spaces

We propose algorithms to take point sets for kernel-based interpolation of functions in reproducing kernel Hilbert spaces (RKHSs) by convex optimization. We consider the case of kernels with the Mercer expansion and propose an algorithm by deriving a second-order cone programming (SOCP) problem that yields $n$ points at one sitting for a given integer $n$. In addition, by modifying the SOCP problem slightly, we propose another sequential algorithm that adds an arbitrary number of new points in each step. Numerical experiments show that in several cases the proposed algorithms compete with the $P$-greedy algorithm, which is known to provide nearly optimal points.Comment: 31 pages. The programs for the numerical computation in this article are available on https://github.com/KeTanakaN/mat_points_interp_rkh