Outer Approximation Approach Research Articles

Robust optimality analysis for linear programming problems with uncertain objective function coefficients: an outer approximation approach

Robust optimality analysis for linear programming problems with uncertain objective function coefficients: an outer approximation approach

Practical algorithms for multivariate rational approximation

We present two approaches for computing rational approximations to multivariate functions, motivated by their effectiveness as surrogate models for high-energy physics (HEP) applications. Our first approach builds on the Stieltjes process to efficiently and robustly compute the coefficients of the rational approximation. Our second approach is based on an optimization formulation that allows us to include structural constraints on the rational approximation (in particular, constraints demanding the absence of singularities), resulting in a semi-infinite optimization problem that we solve using an outer approximation approach. We present results for synthetic and real-life HEP data, and we compare the approximation quality of our approaches with that of traditional polynomial approximations.

Real-Time Energy Management for Smart Homes

This article aims to develop a multiobjective optimization portfolio for real-time energy management in a smart home equipped with battery associated rooftop solar panels, lighting loads, air conditioners, and other smart home appliances. The energy management problem is framed for simultaneous minimization of the monetary energy cost and total dissatisfaction due to regulation in power consumption. The entire optimization portfolio is designed as a time average stochastic problem, which is simplified by the combination of queueing theory and Lyapunov optimization. The revised problem takes form of a mixed integer convex nonlinear programming, which is further solved using outer approximation approach. The proposed real-time home energy management framework needs only the current data regarding the random input parameters like renewable generation, energy price, and aggregated load demand, and does not call for their probabilistic estimation. Case study is carried out on a practical home data to proof efficacy of the proposed strategy and also the simulation outcomes are compared with one of the popular real-time energy management process named greedy algorithm.

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.

A multicut outer-approximation approach for competitive facility location under random utilities

This work concerns the maximum capture facility location problem with random utilities, i.e., the problem of seeking to locate new facilities in a competitive market such that the captured demand of users is maximized, assuming that each individual chooses among all available facilities according to a random utility maximization model. The main challenge lies in the nonlinearity of the objective function. Motivated by the convexity and separable structure of such an objective function, we propose an enhanced implementation of the outer approximation scheme. Our algorithm works in a cutting plane fashion and allows to separate the objective function into a number of sub-functions and create linear cuts for each sub-function at each outer-approximation iteration. We compare our approach with the state-of-the-art method and, for the first time in an extensive way, with other existing nonlinear solvers using three data sets from recent literature. Our experiments show the robustness of our approach, especially on large instances, in terms of both computing time and number instances solved to optimality.

Analytic approximation and differentiability of joint chance constraints

ABSTRACTAn inner–outer approximation approach was recently developed to solve single chance constrained optimization (SCCOPT) problems. In this paper, we extend this approach to address joint chance constrained optimization (JCCOPT) problems. Using an inner–outer approximation, two smooth parametric optimization problems are defined whose feasible sets converge to the feasible set of JCCOPT from inside and outside, respectively. Any optimal solution of the inner approximation problem is a priori feasible to the JCCOPT. As the approximation parameter tends to zero, a subsequence of the solutions of the inner and outer problems, respectively, converge asymptotically to an optimal solution of the JCCOPT. As a main result, the continuous differentiability of the probability function of a joint chance constraint is obtained by examining the uniform convergence of the gradients of the parametric approximations.

Crude-Oil Scheduling Network in Smart Field Under Cyber-Physical System

The smart field has drawn much attention for the field business functions to be tied into a smoothly operating system. However, its operation decisions still cannot meet the ever-increasing multiple data source requirements. This paper proposes a novel smart field cyber-physical system (SF-CPS), which comprises some sensors transmitting real-time sensed information between the marine terminals and the petroleum refinery, and an asset optimization-based decision maker using the pumping schedule to determine the best configuration. From a unique functional unit perspective, the resource allocation of volumes and qualities is implemented with the Dinkelbach method to address this enterprise-wide optimization. Taking advantage of the unloading flows at a low cost, we settle the state variables to the steady process of refinery planning, and then, the next multi-operations sequence follows the tailored outer approximation approach for decomposition to achieve high-cost efficiency. Moreover, the two-phase stochastic scheduling decisions coupled with inventory levels hosted on the SF-CPS platform can cope well with uncertainty in the process between the oil supply and maritime conditions. The experimental results validate the proposed techniques for typical oil and gas resources.

Combinatorial optimal control of semilinear elliptic PDEs

Optimal control problems (OCPs) containing both integrality and partial differential equation (PDE) constraints are very challenging in practice. The most wide-spread solution approach is to first discretize the problem, which results in huge and typically nonconvex mixed-integer optimization problems that can be solved to proven optimality only in very small dimensions. In this paper, we propose a novel outer approximation approach to efficiently solve such OCPs in the case of certain semilinear elliptic PDEs with static integer controls over arbitrary combinatorial structures, where we assume the nonlinear part of the PDE to be non-decreasing and convex. The basic idea is to decompose the OCP into an integer linear programming (ILP) master problem and a subproblem for calculating linear cutting planes. These cutting planes rely on the pointwise concavity or submodularity of the PDE solution with respect to the control variables. The decomposition allows us to use standard solution techniques for ILPs as well as for PDEs. We further benefit from reoptimization strategies for the PDE solution due to the iterative structure of the algorithm. Experimental results show that the new approach is capable of solving the combinatorial OCP of a semilinear Poisson equation with up to 180 binary controls to global optimality within a 5 h time limit. In the case of the screened Poisson equation, which yields semi-infinite integer linear programs, problems with as many as 1400 binary controls are solved.

A multitree approach for global solution of ACOPF problems using piecewise outer approximations

Electricity markets rely on the rapid solution of the optimal power flow (OPF) problem to determine generator power levels and set nodal prices. Traditionally, the OPF problem has been formulated using linearized, approximate models, ignoring nonlinear alternating current (AC) physics. These approaches do not guarantee global optimality or even feasibility in the real ACOPF problem.We introduce an outer-approximation approach to solve the ACOPF problem to global optimality based on alternating solution of upper- and lower-bounding problems. The lower-bounding problem is a piecewise relaxation based on strong second-order cone relaxations of the ACOPF, and these piecewise relaxations are selectively refined at each major iteration through increased variable domain partitioning. Our approach is able to efficiently solve all but one of the test cases considered to an optimality gap below 0.1%. Furthermore, this approach opens the door for global solution of MINLP problems with AC power flow equations.

Joint User Association, Power Allocation, and Throughput Maximization in 5G H-CRAN Networks

In order to get worthwhile gain in coverage and capacity required by fifth generation networks, dense heterogeneous networks are being considered as promising solution. Heterogeneous networks consists of small cells overlaid on macrocells. Introduction of small cells within the coverage of macrocell give birth to problems such as user association, power allocation, and admission control. To address these problems and to maximize overall network throughput heterogeneous networks are integrated with cloud radio access network (CRAN). We formulated a joint user association, admission control, and power allocation problem in heterogeneous CRAN with an objective to maximize overall throughput of network. By examining the structure of the problem, we categorize it under mixed integers nonlinear problems that are generally NP-hard. This paper presents an outer approximation approach (OAA) based on linear programming that simplifies the NP-hard problem, which requires an exhaustive search. OAA ensures $\varepsilon$ -optimal solution with $\varepsilon =10^{-3}$ . OAA converges in a finite number of computations. Effectiveness of proposed algorithm is verified by simulation results.

Joint Admission Control, Mode Selection, and Power Allocation in D2D Communication Systems

Device-to-device (D2D) communications can help in achieving the higher data rate targets in emerging wireless networks. The use of D2D communication imposes certain challenges such as interference with the cellular and D2D users. A well-designed joint admission control, network mode selection, and power allocation technique in a cellular network with D2D capability can improve overall throughput. The proposed technique jointly maximizes the total throughput and number of admitted users in cellular networks under quality-of-service (QoS) and interference constraints. The joint admission control, mode selection, and power allocation problem (JACMSPA) falls into a class of mixed-integer nonlinear constraint optimization problems that are generally NP-hard. Due to the combinatorial nature of the problem, its optimal solution needs exhaustive search of integer variables whose complexity increases exponentially with the number of user pairs. In this paper, we invoke outer approximation approach (OAA)-based linearization technique to solve the JACMSPA. The proposed method gives guaranteed $\varepsilon$ - optimal solution with reasonable computational complexity. Simulation results verify the effectiveness of the proposed approach method.

Polyhedral approximation of ellipsoidal uncertainty sets via extended formulations: a computational case study

Robust optimization is an important technique to immunize optimization problems against data uncertainty. In the case of a linear program and an ellipsoidal uncertainty set, the robust counterpart turns into a second-order cone program. In this work, we investigate the efficiency of linearizing the second-order cone constraints of the latter. This is done using the optimal linear outer-approximation approach by Ben-Tal and Nemirovski (Math Oper Res 26:193–205, 2001) from which we derive an optimal inner approximation of the second-order cone. We examine the performance of this approach on various benchmark sets including portfolio optimization instances as well as (robustified versions of) the MIPLIB and the SNDlib.

Resource Allocation in Hospital Networks Based on Green Cognitive Radios

Pervasive wireless communication and computing can facilitate a variety of e-health applications to communicate patient medical data and information to the doctors/caregivers in an e-Health environment. With the advancement of medical devices and practicing methods the concept of in-cooperating latest communication techniques have taken a new direction by enabling cognitive radio networks in e-Health applications. This paper presents an approach to solve the joint call admission control and power allocation problem in a hospital environment based on green cognitive radio. Specifically, a multi-objective non-convex mixed integer non-linear programming (MINLP) problem of wireless access in an indoor hospital environment is formulated to maximize the number of admitted secondary users and minimize transmit power and carbon dioxide emission. Along with that the throughput requirement of all secondary users and interference constraints for the protected and primary users are also satisfied. In this paper, we invoke outer approximation approach based linearization technique to solve MINLP problem. The proposed method gives guaranteed $$\varepsilon $$? convergence to the optimal solution results with reasonable computational complexity. Simulation results verify the effectiveness of the proposed approach method.

A Cone Constrained Convex Program: Structure and Algorithms

In this paper, we consider the positive semi-definite space tensor cone constrained convex program, its structure and algorithms. We study defining functions, defining sequences and polyhedral outer approximations for this positive semi-definite space tensor cone, give an error bound for the polyhedral outer approximation approach, and thus establish convergence of three polyhedral outer approximation algorithms for solving this problem. We then study some other approaches for solving this structured convex program. These include the conic linear programming approach, the nonsmooth convex program approach and the bi-level program approach. Some numerical examples are presented.

A note on “Berth allocation considering fuel consumption and vessel emissions”

Du et al. [Du, Y., Chen, Q., Quan, X., Long, L., Fung, R.Y.K., 2011. Berth allocation considering fuel consumption and vessel emissions. Transportation Research Part E 47, 1021–1037] dealt with a berth allocation problem incorporating ship’ fuel consumption minimization. To address the difficulty posed by the power function between fuel consumption rate and sailing speed, they formulated a tractable mixed-integer second-order cone programming model. We propose two quadratic outer approximation approaches that can handle general fuel consumption rate functions more efficiently. In the static quadratic outer approximation approach, the approximation lines are generated a priori. In the dynamic quadratic outer approximation approach, the approximation lines are generated dynamically. Numerical experiments demonstrate the advantages of the two approaches.

An outer-approximation approach for information-maximizing sensor selection

This paper addresses information-maximizing sensor selection that determines a set of measurement locations providing the largest entropy reduction in the estimates of the state variables. A new mixed-integer semidefinite program (MISDP) formulation is proposed for this selection under the constraints resulting from communication limitations. This formulation employs binary variables indicating if the corresponding measurement location is selected, and ensures convexity of the objective function and linearity of the constraint functions by exploiting the linear equivalent form of a bilinear term involving binary variables. An outer-approximation algorithm is then developed for the MISDP formulation that obtains the global optimal solution by solving a sequence of mixed-integer linear programs for which reliable solvers are available. Numerical experiments verify the solution optimality and the computational effectiveness of the proposed algorithm by comparing it to branch-and-bound-based approaches with nonlinear programming relaxation. An example of sensor selection to track a moving target is considered to demonstrate the applicability of the proposed method and highlight its ability to handle quadratic constraints.

Generalized Benders’ Decomposition for topology optimization problems

This article considers the non-linear mixed 0---1 optimization problems that appear in topology optimization of load carrying structures. The main objective is to present a Generalized Benders' Decomposition (GBD) method for solving single and multiple load minimum compliance (maximum stiffness) problems with discrete design variables to global optimality. We present the theoretical aspects of the method, including a proof of finite convergence and conditions for obtaining global optimal solutions. The method is also linked to, and compared with, an Outer-Approximation approach and a mixed 0---1 semi definite programming formulation of the considered problem. Several ways to accelerate the method are suggested and an implementation is described. Finally, a set of truss topology optimization problems are numerically solved to global optimality.

Branch-and-Bound Outer Approximation Algorithm for Sum-of-Ratios Fractional Programs

In this article, a branch and-bound outer approximation algorithm is presented for globally solving a sum-of-ratios fractional programming problem. To solve this problem, the algorithm instead solves an equivalent problem that involves minimizing an indefinite quadratic function over a nonempty, compact convex set. This problem is globally solved by a branch-and-bound outer approximation approach that can create several closed-form linear inequality cuts per iteration. In contrast to pure outer approximation techniques, the algorithm does not require computing the new vertices that are created as these cuts are added. Computationally, the main work of the algorithm involves solving a sequence of convex programming problems whose feasible regions are identical to one another except for certain linear constraints. As a result, to solve these problems, an optimal solution to one problem can potentially be used to good effect as a starting solution for the next problem.

On Covering Methods for D.C. Optimization

Covering methods constitute a broad class of algorithms for solving multivariate Global Optimization problems. In this note we show that, when the objective f is d.c. and a d.c. decomposition for f is known, the computational burden usually suffered by multivariate covering methods is significantly reduced. With this we extend to the (non-differentiable) d.c. case the covering method of Breiman and Cutler, showing that it is a particular case of the standard outer approximation approach. Our computational experience shows that this generalization yields not only more flexibility but also faster convergence than the covering method of Breiman-Cutler.

Handling implicit model formulations in MINLP optimization

This paper deals with the handling of implicit model formulations in MINLP optimization and proposes a strategy to cope with occurring difficulties of insufficient information. Process flowsheet simulators are the state-of-the-art tool of chemical engineers. Since these packages often utilize implicit model formulations for physical properties calculation etc., it is necessary to analyze the performance of MINLP algorithms when dealing with such problem representations. The examinations are carried out with the program system PSANO ( Process Synthesis and Network Optimization), where the equation-oriented flowsheeting package SPEEDUP and the Mixed Integer Linear Programming (MILP) solver OSL have been integrated into an extension of the Outer Approximation (OA) approach.