Outer Approximation Research Articles

Feedback control of linear systems with optimal sensor and actuator selection

This article demonstrates a combined feedback control design for linear time-invariant and linear parameter-varying systems and optimal sensors and actuator selection. The combined design problem is systematically constructed as a mixed Boolean semidefinite programming optimization problem. We impose Big-M reformulations to the non-deterministic polynomial-time-hard coupled problem to be solved as a convex optimization problem using the branch and bound algorithm. The combined design of dynamic output feedback control along with optimal actuator selection for a linear time-invariant seismic rejection controller design serves as an application for validation by simulation. In addition, active vibration control of a smart composite plate along with optimal sensor and actuator selection validates the developed approach for linear parameter-varying controller synthesis. On comparing this approach with exhaustive search, it is observed that mixed Boolean semidefinite programming approaches have faster computation time, and comparing with the iterative reweighted ℓ1 norm algorithm and mixed Boolean semidefinite programming using outer approximations, mixed Boolean semidefinite programming yields a global solution.

Tractable inference in credal sentential decision diagrams

Probabilistic sentential decision diagrams are logic circuits where the inputs of disjunctive gates are annotated by probability values. They allow for a compact representation of joint probability mass functions defined over sets of Boolean variables, that are also consistent with the logical constraints defined by the circuit. The probabilities in such a model are usually “learned” from a set of observations. This leads to overconfident and prior-dependent inferences when data are scarce, unreliable or conflicting. In this work, we develop the credal sentential decision diagrams, a generalisation of their probabilistic counterpart that allows for replacing the local probabilities with (so-called credal) sets of mass functions. These models induce a joint credal set over the set of Boolean variables, that sharply assigns probability zero to states inconsistent with the logical constraints. Three inference algorithms are derived for these models. These allow to compute: (i) the lower and upper probabilities of an observation for an arbitrary number of variables; (ii) the lower and upper conditional probabilities for the state of a single variable given an observation; (iii) whether or not all the probabilistic sentential decision diagrams compatible with the credal specification have the same most probable explanation of a given set of variables given an observation of the other variables. These inferences are tractable, as all the three algorithms, based on bottom-up traversal with local linear programming tasks on the disjunctive gates, can be solved in polynomial time with respect to the circuit size. The first algorithm is always exact, while the remaining two might induce a conservative (outer) approximation in the case of multiply connected circuits. A semantics for this approximation together with an auxiliary algorithm able to decide whether or not the result is exact is also provided together with a brute-force characterization of the exact inference in these cases. For a first empirical validation, we consider a simple application based on noisy seven-segment display images. The credal models are observed to properly distinguish between easy and hard-to-detect instances and outperform other generative models not able to cope with logical constraints.

Coordinated Operation of Gas and Electricity Systems for Flexibility Study

The increased interdependencies between electricity and gas systems driven by gas-fired power plants and gas electricity-driven compressors necessitates detailed investigation of such interdependencies, especially in the context of an increased share of renewable energy sources. In this paper, the value of an integrated approach for operating gas and electricity systems is assessed. An outer approximation with equality relaxation (OA/ER) method is used to deal with the optimization class of the mixed-integer non-linear problem of the integrated operation of gas and electricity systems. This method significantly improved the efficiency of the solution algorithm and achieved a nearly 40% reduction in computation time compared to successive linear programming. The value of flexibility technologies, including flexible gas compressors, demand-side response, battery storage, and power-to-gas, is quantified in the operation of integrated gas and electricity systems in GB 2030 energy scenarios for different renewable generation penetration levels. The modeling demonstrates that the flexibility options will enable significant cost savings in the annual operational costs of gas and electricity systems (up to 21%). On the other hand, the analysis carried out indicates that deployment of flexibility technologies appropriately supports the interaction between gas and electricity systems.

Accelerated hybrid methods for solving pseudomonotone equilibrium problems

In this paper, we introduce some new accelerated hybrid algorithms for solving a pseudomonotone equilibrium problem with a Lipschitz-type condition in a Hilbert space. The algorithms are constructed around the extragradient method, the inertial technique, the hybrid (or outer approximation) method, and the shrinking projection method. The algorithms are designed to work either with or without the prior knowledge of the Lipschitz-type constants of bifunction. Theorems of strong convergence are established under mild conditions. The results in this paper generalize, extend, and improve some known results in the field. Finally, several of numerical experiments are performed to support the obtained theoretical results.

Parallel modified methods for pseudomonotone equilibrium problems and fixed point problems for quasi-nonexpansive mappings

The paper considers the problem of finding common solutions of a system of pseudomonotone equilibrium problems and fixed point problems for quasi-nonexpansive mappings. The problem covers various mathematical models of convex feasibility problems and the problems whose constraints are expressed by the intersection of fixed point sets of mappings. The main purpose of the paper is to design and improve computations over each step and weaken several assumptions imposed on bifunctions and mappings. Two parallel algorithms for finding of a particular solution of the problem are proposed in Hilbert spaces where each subproblem in the family can be computed simultaneously. The first one is a modified hybrid method which combines three methods including the generalized gradient-like projection method, the Mann’s iteration and the hybrid (outer approximation) method. This algorithm improves the hybrid extragradient method at each computational step where only one optimization problem is solved for each equilibrium subproblem in the family and the hybrid step does not deal with the feasible set of the considered problem. The strong convergence of the algorithm comes from the hybrid method under the Lipschitz-type condition of bifunctions. The second algorithm is a viscosity-like method with a linesearch procedure that aims to avoid the Lipschitz-type condition imposed on bifunctions. With the incorporated viscosity technique, the algorithm also provides strong convergence. Several numerical experiments are performed to illustrate the efficiency of the proposed algorithms and also to compare them with known parallel hybrid extragradient methods.

Solving a production-routing problem with price-dependent demand using an outer approximation method

A production-routing problem with price-dependent demand (PRP-PD) is studied in this paper. Demand follows a general convex, differentiable, continuous and strictly decreasing function in price. The problem is modeled as a mixed integer nonlinear program (MINLP). Two Outer Approximation (OA) based algorithms are developed to solve the PRP-PD. The efficiency of the proposed algorithms in comparison with commercial MINLP solvers is demonstrated. The computational results show that our basic OA algorithm outperforms the commercial solvers both in solution quality and in computational time aspects. On the other hand, our extended (two-phase) OA algorithm provides near-optimal solutions very efficiently, especially for large problem instances. These findings prevail both for linear and for nonlinear demand functions. Additional sensitivity analyses are conducted to investigate the impact of different problem parameters on the optimal solution. The results show that the manufacturer should give higher priority to the retailer who has lower price sensitivity and who is closer to the manufacturer. Another takeaway is that a larger market size and a lower price sensitivity lead to more profit.

A monotonic optimization approach for solving strictly quasiconvex multiobjective programming problems

In this article, we use the monotonic optimization approach to propose an outcome-space outer approximation by copolyblocks for solving strictly quasiconvex multiobjective programming problems and especially in the case that the objective functions are nonlinear fractional. After the algorithm is terminated, with any given tolerance, we obtain an approximation of the weakly efficient solution set, that contains the whole weakly efficient solution set of the problem. The algorithm is proved to be convergent and it is suitable to be implemented in parallel using standard convex programming tools. Some computational experiments are reported to show the accuracy and efficiency of the proposed algorithm.

Efficient Singularity-Free Workspace Approximations Using Sum-of-Squares Programming

Abstract In this paper, we provide a general framework to determine inner and outer approximations to the singularity-free workspace of fully actuated robotic manipulators, subject to Type-I and Type-II singularities. This framework utilizes the sum-of-squares optimization technique, which is numerically implemented by semidefinite programming. In order to apply the sum-of-squares optimization technique, we convert the trigonometric functions in the kinematics of the manipulator to polynomial functions with an additional constraint. We define two quadratic forms, describing two ellipsoids, whose volumes are optimized to yield inner and outer approximations of the singularity-free workspace.

A new solution method for a class of large dimension rank-two nonconvex programs

Abstract Low-rank problems are nonlinear minimization problems in which the objective function, by means of a suitable linear transformation of the variables, depends on very few variables. These problems often arise in quantitative management science applications, for example, in location models, transportation problems, production planning, data envelopment analysis and multiobjective programs. They are usually approached by means of outer approximation, branch and bound, branch and select and optimal level solution methods. The paper studies, from both a theoretical and an algorithmic point of view, a class of large-dimension rank-two nonconvex problems having a polyhedral feasible region and $f(x)=\phi (c^Tx+c_0,d^Tx+d_0)$ as the objective function. The proposed solution algorithm unifies a new partitioning method, an outer approximation approach and a mixed method. The results of a computational test are provided to compare these three approaches with the optimal level solutions method. In particular, the new partitioning method performs very well in solving large problems.

Energy-Efficient Resource Allocation for High-Rate Underlay D2D Communications With Statistical CSI: A One-to-Many Strategy

For the prominent superiorities in energy and spectrum efficiency, Device-to-Device (D2D) communication has become a hot topic among the 5th generation (5G) technologies. However, the widely adopted mode that one or more D2D links reuse one cellular user‘s uplink spectrum may lead to severe limitation on D2D communication rate, especially when cellular user's uplink spectrum is very limited. In this paper, we will face the high rate requirements of D2D pairs (DPs), with the help of carrier aggregation technology, each DP can reuse the uplink spectrum of multiple cellular users when needed. More practically, we consider that only statistical channel state information (CSI) of certain communication links is available here. Then, we formulate the problem to minimize the total power consumption of the mobile devices to obtain an energy-efficient resource allocation result, including spectrum and power allocation. Meanwhile, the quality of service (QoS) requirements of cellular and high-rate D2D communications are both ensured. The formulated problem is mathematically a non-convex mixed-integer nonlinear programming (MINLP) problem, which is NP-hard. To solve it, we first tighten the constraints and deploy transformation methods on it to make it convex. Then, we propose a two-layer algorithm, the independent-power greedy-based outer approximation (IPGOA), to solve the transformed problem. Besides, to handle the involved uniform power allocation circumstance in the carrier aggregation process, a uniform-power GOA (UPGOA) algorithm, which can be regarded as a simplified version of IPGOA, is also proposed. The simulation results show that in different scenarios, our proposed scheme is approximately optimal.

Resource Allocation for Non-Orthogonal Multiple Access-Enabled Fog Radio Access Networks

Non-orthogonal multiple access (NOMA) has been considered as a promising communication technology to enhance the spectral efficiency and support massive connections in fog radio access networks (F-RANs). In this paper, with the aim of maximizing the weighted sum rate while taking co-channel interference into consideration, a joint resource block (RB) and power allocation problem is formulated. To solve this problem, we first propose the optimal resource allocation scheme. Specifically, the monotonic optimization is applied and an outer polyblock approximation algorithm is proposed to get the global optimal solution. In order to reduce the computational complexity, we then propose the suboptimal resource allocation scheme. In particular, the original problem is decomposed into separated RB and power allocation problems. The RB allocation problem is modeled as a many-to-one matching game and a modified swap-enabled matching algorithm is proposed. The power allocation problem is converted into a convex form through some approximations and solved by a successive convex approximation algorithm. Simulation results demonstrate that the suboptimal scheme can achieve almost the same performance as the optimal scheme, while requiring much less computational complexity. In addition, the superiority of NOMA-enabled F-RANs over the conventional OMA-enabled F-RANs is verified.

Outer Approximation Methods for Solving Variational Inequalities Defined over the Solution Set of a Split Convex Feasibility Problem

We study variational inequalities which are governed by a strongly monotone and Lipschitz continuous operator F over a closed and convex set S. We assume that is the nonempty solution set of a (multiple-set) split convex feasibility problem, where C and Q are both closed and convex subsets of two real Hilbert spaces and respectively, and the operator A acting between them is linear. We consider a modification of the gradient projection method the main idea of which is to replace at each step the metric projection onto S by another metric projection onto a half-space which contains S. We propose three variants of a method for constructing the above-mentioned half-spaces by employing the multiple-set and the split structure of the set S. For the split part we make use of the Landweber transform.

Inner approximation algorithm for solving linear multiobjective optimization problems

ABSTRACT Benson's outer approximation algorithm and its variants are the most frequently used methods for solving linear multiobjective optimization problems. These algorithms have two intertwined parts: single-objective linear optimization on one hand, and a combinatorial part closely related to vertex enumeration on the other. Their separation provides a deeper insight into Benson's algorithm, and points toward a dual approach. Two skeletal algorithms are defined which focus on the combinatorial part. Using different single-objective optimization problems yields different algorithms, such as a sequential convex hull algorithm, another version of Benson's algorithm with the theoretically best possible iteration count, and the dual algorithm of Ehrgott et al. [A dual variant of Benson's ‘outer approximation algorithm’ for multiple objective linear programming. J Glob Optim. 2012;52:757–778]. The implemented version is well suited to handle highly degenerate problems where there are many linear dependencies among the constraints. On problems with 10 or more objectives, it shows a significant increase in efficiency compared to Bensolve – due to the reduced number of iterations and the improved combinatorial handling.

An Optimized Covering Spheroids by Spheres

Covering spheroids (ellipsoids of revolution) by different spheres is studied. The research is motivated by packing non-spherical particles arising in natural sciences, e.g., in powder technologies. The concept of an ε -cover is introduced as an outer multi-spherical approximation of the spheroid with the proximity ε . A fast heuristic algorithm is proposed to construct an optimized ε -cover giving a reasonable balance between the value of the proximity parameter ε and the number of spheres used. Computational results are provided to demonstrate the efficiency of the approach.

Reliability assessment of composite generation and transmission expansion planning incorporating renewable energy sources

Modern power system exploitation of large-scale Renewable Energy Sources (RESs) is prevalent due to the RES effect of reducing the heavy dependence on fossil fuels and the resultant power quality improvement. Therefore, this paper investigates the reliability assessment of a power system with integration of large-scale RESs with emphasis on their impacts on the expansion planning procedure. A multi-state Markov model is proposed to analyze the availability and reliability of the stochastic nature of RESs and High Voltage Direct Current transmission links. A Mixed Integer Nonlinear Programming model is developed to minimize the total investment and operation cost of the generating units and the cost associated with power outage as well as improving power system reliability. An Outer Approximation Algorithm, a powerful solver embedded in the Advanced Interactive Multidimensional Modeling System, was employed in solving the proposed model and implemented using the IEEE 6- and 24-bus Roy Billinton Test System to demonstrate its effectiveness. The results obtained from the simulation reveal that the proposed model performance achieves effective reliability and an enhanced voltage profile.

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.

The decomposition-based outer approximation algorithm for convex mixed-integer nonlinear programming

This paper presents a new two-phase method for solving convex mixed-integer nonlinear programming (MINLP) problems, called Decomposition-based Outer Approximation Algorithm (DECOA). In the first phase, a sequence of linear integer relaxed sub-problems (LP phase) is solved in order to rapidly generate a good linear relaxation of the original MINLP problem. In the second phase, the algorithm solves a sequence of mixed integer linear programming sub-problems (MIP phase). In both phases the outer approximation is improved iteratively by adding new supporting hyperplanes by solving many easier sub-problems in parallel. DECOA is implemented as a part of Decogo (Decomposition-based Global Optimizer), a parallel decomposition-based MINLP solver implemented in Python and Pyomo. Preliminary numerical results based on 70 convex MINLP instances up to 2700 variables show that due to the generated cuts in the LP phase, on average only 2–3 MIP problems have to be solved in the MIP phase.

Outer approximation for integer nonlinear programs via decision diagrams

As an alternative to traditional integer programming (IP), decision diagrams (DDs) provide a new solution technology for discrete problems exploiting their combinatorial structure and dynamic programming representation. While the literature mainly focuses on the competitive aspects of DDs as a stand-alone solver, we investigate their complementary role by introducing IP techniques that can be derived from DDs and used in conjunction with IP to enhance the overall performance. This perspective allows for studying problems with more general structure than those typically modeled via recursive formulations. In particular, we develop linear programming and subgradient-type methods to generate valid inequalities for the convex hull of the feasible region described by DDs. For convex IPs, these cutting planes dominate the so-called linearized cuts used in the outer approximation framework. These cutting planes can also be derived for nonconvex IPs, which leads to a generalization of the outer approximation framework. Computational experiments show significant optimality gap improvement for integer nonlinear programs over the traditional cutting plane methods employed in the state-of-the-art solvers.

Two-Stage Stochastic Choice Modeling Approach for Electric Vehicle Charging Station Network Design in Urban Communities

Electric vehicles (EVs) provide a cleaner alternative that not only reduces greenhouse gas emissions but also improves air quality and reduces noise pollution. The consumer market for electrical vehicles is growing very rapidly. Designing a network with adequate capacity and types of public charging stations is a challenge that needs to be addressed to support the current trend in the EV market. In this research, we propose a choice modeling approach embedded in a two-stage stochastic programming model to determine the optimal layout and types of EV supply equipment for a community while considering randomness in demand and drivers' behaviors. Some of the key random data parameters considered in this study are: EV's dwell time at parking {\sv location}, battery's state of charge, distance from home, willingness to walk, drivers' arrival patterns, and traffic on weekdays and weekends. The two-stage model uses the sample average approximation method, which asymptotically converges to an optimal solution. To address the computational challenges for large-scale instances, we propose an outer approximation decomposition algorithm. We conduct extensive computational experiments to quantify the efficacy of the proposed approach. In addition, we present the results and a sensitivity analysis for a case study based on publicly available data sources.

3D UAV Trajectory and Communication Design for Simultaneous Uplink and Downlink Transmission

In this paper, we investigate the unmanned aerial vehicle (UAV)-aided simultaneous uplink and downlink transmission networks, where one UAV acting as a disseminator is connected to multiple access points (AP), and the other UAV acting as a base station (BS) collects data from numerous sensor nodes (SNs). The goal of this paper is to maximize the system throughput by jointly optimizing the 3D UAV trajectory, communication scheduling, and UAV-AP/SN transmit power. We first consider a special case where the UAV-BS and UAV-AP trajectories are pre-determined. Although the resulting problem is an integer and non-convex optimization problem, a globally optimal solution is obtained by applying the polyblock outer approximation (POA) method based on the problem's hidden monotonic structure. Subsequently, for the general case considering the 3D UAV trajectory optimization, an efficient iterative algorithm is proposed to alternately optimize the divided sub-problems based on the successive convex approximation (SCA) technique. Numerical results demonstrate that the proposed design is able to achieve significant system throughput gain over the benchmarks. In addition, the SCA-based method can achieve nearly the same performance as the POA-based method with much lower computational complexity.