Mixed-integer Convex Optimization Research Articles

Information complexity of mixed-integer convex optimization

Information complexity of mixed-integer convex optimization

Power Losses Reduction in Three-Phase Unbalanced Distribution Networks Using a Convex Optimization Model in the Complex Domain

This article presents a solution methodology to minimize power losses in three-phase unbalanced distribution networks. This approach involved an efficient complex-domain model that is categorized under mixed-integer convex optimization. The methodology employed consisted of efficient load rotation at each constant power node via a three-phase rotation matrix that allows defining each load connection to minimize the expected power imbalance at the terminals of the substation, as well as the total grid power losses, and improve voltage profile performance at each system phase. The load imbalance, expressed as a percentage, can be defined as a function of the active, reactive, or apparent power. In addition, considering the complex-domain representation of three-phase electrical networks under steady-state conditions, a mixed-integer convex model was formulated to reduce the power imbalances. With the purpose of determining the initial and final power losses of these distribution systems, the successive approximations method was employed to address the three-phase power flow problem. As a result, numerical validations in the IEEE 25-bus system and a 35-node three-phase feeder showed that the final active power losses vary depending on the objective function analyzed. Therefore, for the test feeders studied, it is necessary to evaluate each objective function, with the aim of finding the one that yields the best numerical results. Power losses reductions of about 3.8056 % and 6.8652 % were obtained for both test feeders via the proposed optimization methodology. All numerical validations were performed in the Julia programming environment, using the JuMP optimization tool and the HiGHS solver.

A Solver for Multiobjective Mixed-Integer Convex and Nonconvex Optimization

AbstractThis paper proposes a general framework for solving multiobjective nonconvex optimization problems, i.e., optimization problems in which multiple objective functions have to be optimized simultaneously. Thereby, the nonconvexity might come from the objective or constraint functions, or from integrality conditions for some of the variables. In particular, multiobjective mixed-integer convex and nonconvex optimization problems are covered and form the motivation of our studies. The presented algorithm is based on a branch-and-bound method in the pre-image space, a technique which was already successfully applied for continuous nonconvex multiobjective optimization. However, extending this method to the mixed-integer setting is not straightforward, in particular with regard to convergence results. More precisely, new branching rules and lower bounding procedures are needed to obtain an algorithm that is practically applicable and convergent for multiobjective mixed-integer optimization problems. Corresponding results are a main contribution of this paper. What is more, for improving the performance of this new branch-and-bound method we enhance it with two types of cuts in the image space which are based on ideas from multiobjective mixed-integer convex optimization. Those combine continuous convex relaxations with adaptive cuts for the convex hull of the mixed-integer image set, derived from supporting hyperplanes to the relaxed sets. Based on the above ingredients, the paper provides a new multiobjective mixed-integer solver for convex problems with a stopping criterion purely in the image space. What is more, for the first time a solver for multiobjective mixed-integer nonconvex optimization is presented. We provide the results of numerical tests for the new algorithm. Where possible, we compare it with existing procedures.

A Prescriptive Machine Learning Approach to Mixed-Integer Convex Optimization

We introduce a prescriptive machine learning approach to speed up the process of solving mixed-integer convex optimization (MICO) problems. We solve multiple optimization instances and train a machine learning model in advance, which we use to solve new instances. Previous works have shown that the predictions of classification algorithms enable us to solve optimization problems much faster than commercial solvers. What distinguishes this paper from the previous work is that we use a prescriptive algorithm, Optimal Policy Trees (OPT), instead of classification algorithms. Whereas classification algorithms aim to predict the correct label and consider all other labels equally undesirable, a prescriptive approach takes into account all the available decision options and their counterfactuals. We first introduce an algorithm that is purely based on OPT and also its extension. We compare their performance with Optimal Classification Trees (OCT) on various MICO problems. Test problems include transportation optimization, portfolio optimization, facility location, and hybrid vehicle control. We also experiment on real-world instances taken from the Mixed Integer Programming Library. OPT-based methods have a significant edge on finding feasible solutions, whereas OCT-based methods have a slight edge on the degree of suboptimality. The proposed extension of the pure OPT algorithm improves on the suboptimality of the solutions the algorithm produces. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: The research was funded in part by a grant from OCP to MIT.

A hybrid patch decomposition approach to compute an enclosure for multi-objective mixed-integer convex optimization problems

AbstractIn multi-objective mixed-integer convex optimization, multiple convex objective functions need to be optimized simultaneously while some of the variables are restricted to take integer values. In this paper, we present a new algorithm to compute an enclosure of the nondominated set of such optimization problems. More precisely, we decompose the multi-objective mixed-integer convex optimization problem into several multi-objective continuous convex optimization problems, which we refer to as patches. We then dynamically compute and improve coverages of the nondominated sets of those patches to finally combine them to obtain an enclosure of the nondominated set of the multi-objective mixed-integer convex optimization problem. Additionally, we introduce a mechanism to reduce the number of patches that need to be considered in total. Our new algorithm is the first of its kind and guaranteed to return an enclosure of prescribed quality within a finite number of iterations. For selected numerical test instances we compare our new criterion space based approach to other algorithms from the literature and show that much larger instances can be solved with our new algorithm.

A Planning Method for Synchronous Condensers in Weak Grids Using Semi-Definite Optimization

Synchronous Condensers (SynCons) offer voltage regulation, inertia, and fault current contribution to solve the challenges of voltage and frequency instability introduced by the high penetration of renewable energy resources (RERs). However, the cost of installation and operation of a SynCon is noticeable. Furthermore, the SynCons installation can take more than a year, which means the number, size, and placement of the SynCons must be selected optimally. This paper proposes a method to formulate the optimal sizing and allocation of the SynCons via mixed-integer convex optimization. The optimization procedure aims to minimize the cost of SynCons installation, maintenance, and operation while maintaining a certain Short Circuit Ratio (SCR) at points of connection (PoCs). To ensure the SCR at the PoCs is greater than a certain threshold, the SCRs are formulated as constraints in the proposed optimization procedure. The modified IEEE 39-bus system is used to evaluate the proposed optimization method. The performance of the system with optimal SynCons is verified by electromagnetic transient time-domain simulation using PSCAD/EMTDC software. The simulation results show that the system with the optimal installation of SynCons performs better than the system without the SynCons or with a non-optimal installation of SynCons. With the optimized system, all renewable energy resources are stable after contingencies.

Complexity of optimizing over the integers

In the first part of this paper, we present a unified framework for analyzing the algorithmic complexity of any optimization problem, whether it be continuous or discrete in nature. This helps to formalize notions like “input”, “size” and “complexity” in the context of general mathematical optimization, avoiding context dependent definitions which is one of the sources of difference in the treatment of complexity within continuous and discrete optimization. In the second part of the paper, we employ the language developed in the first part to study information theoretic and algorithmic complexity of mixed-integer convex optimization, which contains as a special case continuous convex optimization on the one hand and pure integer optimization on the other. We strive for the maximum possible generality in our exposition. We hope that this paper contains material that both continuous optimizers and discrete optimizers find new and interesting, even though almost all of the material presented is common knowledge in one or the other community. We see the main merit of this paper as bringing together all of this information under one unifying umbrella with the hope that this will act as yet another catalyst for more interaction across the continuous-discrete divide. In fact, our motivation behind Part I of the paper is to provide a common language for both communities.

Mixed-Integer Convex Representability

Motivated by recent advances in solution methods for mixed-integer convex optimization (MICP), we study the fundamental and open question of which sets can be represented exactly as feasible regions of MICP problems. We establish several results in this direction, including the first complete characterization for the mixed-binary case and a simple necessary condition for the general case. We use the latter to derive the first nonrepresentability results for various nonconvex sets, such as the set of rank-1 matrices and the set of prime numbers. Finally, in correspondence with the seminal work on mixed-integer linear representability by Jeroslow and Lowe, we study the representability question under rationality assumptions. Under these rationality assumptions, we establish that representable sets obey strong regularity properties, such as periodicity, and we provide a complete characterization of representable subsets of the natural numbers and of representable compact sets. Interestingly, in the case of subsets of natural numbers, our results provide a clear separation between the mathematical modeling power of mixed-integer linear and mixed-integer convex optimization. In the case of compact sets, our results imply that using unbounded integer variables is necessary only for modeling unbounded sets.

Optimal Demand Reconfiguration in Three-Phase Distribution Grids Using an MI-Convex Model

The problem of the optimal load redistribution in electrical three-phase medium-voltage grids is addressed in this research from the point of view of mixed-integer convex optimization. The mathematical formulation of the load redistribution problem is developed in terminals of the distribution node by accumulating all active and reactive power loads per phase. These loads are used to propose an objective function in terms of minimization of the average unbalanced (asymmetry) grade of the network with respect to the ideal mean consumption per-phase. The objective function is defined as the l1-norm which is a convex function. As the constraints consider the binary nature of the decision variable, each node is conformed by a 3×3 matrix where each row and column have to sum 1, and two equations associated with the load redistribution at each phase for each of the network nodes. Numerical results demonstrate the efficiency of the proposed mixed-integer convex model to equilibrate the power consumption per phase in regards with the ideal value in three different test feeders, which are composed of 4, 15, and 37 buses, respectively.

Exact minimization of the energy losses and the CO2 emissions in isolated DC distribution networks using PV sources

This paper addresses the optimal location and sizing of photovoltaic (PV) sources in isolated direct current (DC) electrical networks, considering time-varying load and renewable generation curves. The mathematical formulation of this problem corresponds to mixed-integer nonlinear programming (MINLP), which is reformulated via mixed-integer convex optimization: This ensures the global optimum solving the resulting optimization model via branch & bound and interior-point methods. The main idea of including PV sources in the DC grid is to minimize the daily energy losses and greenhouse emissions produced by diesel generators in isolated areas. The GAMS package is employed to solve the MINLP model, using mixed and integer variables; also, the CVX and MOSEK solvers are used to obtain solutions from the proposed mixed-integer convex model in the MATLAB. Numerical results demonstrate important reductions in the daily energy losses and the harmful gas emissions when PV sources are optimally integrated into DC grid.

Tax-Aware Portfolio Construction via Convex Optimization

We describe an optimization-based tax-aware portfolio construction method that adds tax liability to standard Markowitz-based portfolio construction. Our method produces a trade list that specifies the number of shares to buy of each asset and the number of shares to sell from each tax lot held. To avoid wash sales (in which some realized capital losses are disallowed), we assume that we trade monthly and cannot simultaneously buy and sell the same asset. The tax-aware portfolio construction problem is not convex, but it becomes convex when we specify, for each asset, whether we buy or sell it. It can be solved using standard mixed-integer convex optimization methods at the cost of very long solve times for some problem instances. We present a custom convex relaxation of the problem that borrows curvature from the risk model. This relaxation can provide a good approximation of the true tax liability, while greatly enhancing computational tractability. This method requires the solution of only two convex optimization problems: the first determines whether we buy or sell each asset, and the second generates the final trade list. In our numerical experiments, our method almost always solves the nonconvex problem to optimality, and when it does not, it produces a trade list very close to optimal. Backtests show that the performance of our method is indistinguishable from that obtained using a globally optimal solution, but with significantly reduced computational effort.

A model predictive approach for online mobile manipulation of non-holonomic objects using learned dynamics

Assistive robots designed for physical interaction with objects will play an important role in assisting with mobility and fall prevention in healthcare facilities. Autonomous mobile manipulation presents a hurdle prior to safely using robots in real-life applications. In this article, we introduce a mobile manipulation framework based on model predictive control using learned dynamics models of objects. We focus on the specific problem of manipulating legged objects such as those commonly found in healthcare environments and personal dwellings (e.g., walkers, tables, chairs). We describe a probabilistic method for autonomous learning of an approximate dynamics model for these objects. In this method, we learn dynamic parameters using a small dataset consisting of force and motion data from interactions between the robot and object. Moreover, we account for multiple manipulation strategies by formulating manipulation planning as a mixed-integer convex optimization. The proposed framework considers the hybrid control system composed of (i) choosing which leg to grasp and (ii) control of continuous applied forces for manipulation. We formalize our algorithm based on model predictive control to compensate for modeling errors and find an optimal path to manipulate the object from one configuration to another. We present results for several objects with various wheel configurations. Simulation and physical experiments show that the obtained dynamics models are sufficiently accurate for safe and collision-free manipulation. When combined with the proposed manipulation planning algorithm, the robot successfully moves the object to the desired pose while avoiding any collision.

Generating Feasible Points for Mixed-Integer Convex Optimization Problems by Inner Parallel Cuts

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 14 February 2020Accepted: 17 June 2021Published online: 28 September 2021Keywordsconvex MINLP, feasible rounding, granularity, inner parallel set, cutting planes, extended cutting plane algorithm, global error bound, outer approximationAMS Subject Headings90C11, 90C10, 90C31, 90C30Publication DataISSN (print): 1052-6234ISSN (online): 1095-7189Publisher: Society for Industrial and Applied MathematicsCODEN: sjope8

The voice of optimization

We introduce the idea that using optimal classification trees (OCTs) and optimal classification trees with-hyperplanes (OCT-Hs), interpretable machine learning algorithms developed by Bertsimas and Dunn (Mach Learn 106(7):1039–1082, 2017), we are able to obtain insight on the strategy behind the optimal solution in continuous and mixed-integer convex optimization problem as a function of key parameters that affect the problem. In this way, optimization is not a black box anymore. Instead, we redefine optimization as a multiclass classification problem where the predictor gives insights on the logic behind the optimal solution. In other words, OCTs and OCT-Hs give optimization a voice. We show on several realistic examples that the accuracy behind our method is in the 90–100% range, while even when the predictions are not correct, the degree of suboptimality or infeasibility is very low. We compare optimal strategy predictions of OCTs and OCT-Hs and feedforward neural networks (NNs) and conclude that the performance of OCT-Hs and NNs is comparable. OCTs are somewhat weaker but often competitive. Therefore, our approach provides a novel insightful understanding of optimal strategies to solve a broad class of continuous and mixed-integer optimization problems.

Outer approximation with conic certificates for mixed-integer convex problems

A mixed-integer convex (MI-convex) optimization problem is one that becomes convex when all integrality constraints are relaxed. We present a branch-and-bound LP outer approximation algorithm for an MI-convex problem transformed to MI-conic form. The polyhedral relaxations are refined with $\mathcal{K}^*$ cuts derived from conic certificates for continuous primal-dual conic subproblems. Under the assumption that all subproblems are well-posed, the algorithm detects infeasibility or unboundedness or returns an optimal solution in finite time. Using properties of the conic certificates, we show that the $\mathcal{K}^*$ cuts imply certain practically-relevant guarantees about the quality of the polyhedral relaxations, and demonstrate how to maintain helpful guarantees when the LP solver uses a positive feasibility tolerance. We discuss how to disaggregate $\mathcal{K}^*$ cuts in order to tighten the polyhedral relaxations and thereby improve the speed of convergence, and propose fast heuristic methods of obtaining useful $\mathcal{K}^*$ cuts. Our new open source MI-conic solver Pajarito (http://github.com/JuliaOpt/Pajarito.jl) uses an external mixed-integer linear (MILP) solver to manage the search tree and an external continuous conic solver for subproblems. Benchmarking on a library of mixed-integer second-order cone (MISOCP) problems, we find that Pajarito greatly outperforms Bonmin (the leading open source alternative) and is competitive with CPLEX's specialized MISOCP algorithm. We demonstrate the robustness of Pajarito by solving diverse MI-conic problems involving mixtures of positive semidefinite, second-order, and exponential cones, and provide evidence for the practical value of our analyses and enhancements of $\mathcal{K}^*$ cuts.

Global inverse kinematics via mixed-integer convex optimization

In this paper, we present a novel formulation of the inverse kinematics (IK) problem with generic constraints as a mixed-integer convex optimization program. The proposed approach can solve the IK problem globally with generic task space constraints: a major improvement over existing approaches, which either solve the problem in only a local neighborhood of the user initial guess through nonlinear non-convex optimization, or address only a limited set of kinematics constraints. Specifically, we propose a mixed-integer convex relaxation of non-convex rotation constraints, and apply this relaxation on the IK problem. Our formulation can detect if an instance of the IK problem is globally infeasible, or produce an approximate solution when it is feasible. We show results on a seven-joint arm grasping objects in a cluttered environment, an 18-degree-of-freedom quadruped standing on stepping stones, and a parallel Stewart platform. Moreover, we show that our approach can find a collision free path for a gripper in a cluttered environment, or certify such a path does not exist. We also compare our approach against the analytical approach for a six-joint manipulator. The open-source code is available at http://drake.mit.edu .

Optimal trajectories and normal load analysis of hypersonic glide vehicles via convex optimization

Hypersonic trajectory optimization has been intensively investigated through different approaches; however, the normal-load-optimal entry problems were barely studied and reported in the literature. Finding the optimal trajectories with maximum or minimum peak normal load is essential to evaluate the maneuverability and structural strength of the vehicle. In this paper, both the maximum and minimum peak-normal-load entry trajectories are explored using convex optimization. Based on the previous work, the maximum-peak-normal-load entry problem is firstly addressed by a Big-M method and a line-search approach. Through successive relaxations, the nonconvex discrete-event optimal control problem associated with maximum-peak-normal-load entry is transformed into a sequence of mixed-integer convex optimization problems. Then, a line-search technique is introduced to improve the convergence of the proposed method. Additionally, a sequential convex programming method is designed to solve the minimum-peak-normal-load entry problem to comprehensively analyze the normal load during the entry flight. There are efficient solvers that can solve each relaxed convex subproblem with a global optimum if the feasible set of the subproblem is nonempty. The convergence and accuracy of the proposed methodologies are demonstrated by numerical simulations, and the feasibility of the converged solutions is discussed based on an entry-corridor approach.

A Joint Vehicle Routing and Speed Optimization Problem

Classic vehicle routing models usually treat fuel cost as input data, but fuel consumption heavily depends on the travel speed, which leads to the study of optimizing speeds over a route to improve fuel efficiency. In this paper, we propose a joint vehicle routing and speed optimization problem to minimize the total operating cost including fuel cost. The only assumption made on the dependence between fuel cost and travel speed is that it is a strictly convex differentiable function. This problem is very challenging, with medium-sized instances already difficult for a general mixed-integer convex optimization solver. We propose a novel set-partitioning formulation and a branch-cut-and-price algorithm to solve this problem. We introduce new dominance rules for the labeling algorithm so that the pricing problem can be solved efficiently. Our algorithm clearly outperforms the off-the-shelf optimization solver, and is able to solve some benchmark instances to optimality for the first time. The online supplement is available at https://doi.org/10.1287/ijoc.2018.0810 .

Polyhedral approximation in mixed-integer convex optimization

Generalizing both mixed-integer linear optimization and convex optimization, mixed-integer convex optimization possesses broad modeling power but has seen relatively few advances in general-purpose solvers in recent years. In this paper, we intend to provide a broadly accessible introduction to our recent work in developing algorithms and software for this problem class. Our approach is based on constructing polyhedral outer approximations of the convex constraints, resulting in a global solution by solving a finite number of mixed-integer linear and continuous convex subproblems. The key advance we present is to strengthen the polyhedral approximations by constructing them in a higher-dimensional space. In order to automate this extended formulation we rely on the algebraic modeling technique of disciplined convex programming (DCP), and for generality and ease of implementation we use conic representations of the convex constraints. Although our framework requires a manual translation of existing models into DCP form, after performing this transformation on the MINLPLIB2 benchmark library we were able to solve a number of unsolved instances and on many other instances achieve superior performance compared with state-of-the-art solvers like Bonmin, SCIP, and Artelys Knitro.

Rate-sensitive CoMP beamforming plus sensible snooze for energy-QoE tradeoff in cognitive small cell networks

Small cells, specific femtocells aim to improve throughput and customer experience indoors, which prospectively develop into an indispensable part to complement macrocells. Since network operators progressively consider ecological and economical sustainable development, energy-saving and interference-free have grown to be crucial challenges for “green” femtocells, especially under ultra-dense deployment. To this end, we introduce coordinated multi-point (CoMP) femto-base stations (FBSs) to exploit intra-femto interference and multiple-input single-output beamforming in cognitive underlay to mitigate cross-tier interference, integrated with another key effort to best conserve overall FBSs power. It is formulated as a quasi mixed-integer convex optimization problem, decomposed and carried out separately via innovative transformations. A novel FBSs CoMP-selecting scheme, sensitive to femto-user traffic rate, is proposed to rapidly obtain the effective solution to fundamental CoMP beamforming issue. Furthermore, a policy of diminishing the CoMP set by deactivating feasible no-load FBSs to snooze, is proposed to actually minimize entire power consumption, wherein FBSs static, dynamic and backhauling power is jointly evaluated by accordingly proposed two hierarchical iterative algorithms. Network-level simulations under practical configurations, in contrast to prior work, demonstrate the proposals can achieve a proper tradeoff between retaining significant energy-efficient FBSs operation and assuring preferable quality of consumers experience.