Solution Of Mixed Integer Nonlinear Programming Research Articles

A method for power flow calculation and optimal dispatch of gas–thermal–electricity multi energy system considering unit commitment

The optimal scheduling of integrated energy system considering unit commitment is a mixed integer nonlinear programming (MINLP) problem with 0-1 binary variables, and the efficient solution of MINLP has always been a research hotspot. In this paper, a power flow calculation method of gas thermal power multi energy coupling system is proposed. Considering the output of coal-fired, gas-fired and new energy units, a simplified unit commitment calculation model is established, which can greatly reduce the number of binary variables and constraints. At the same time, an optimal scheduling model with minimum economic cost and minimum carbon emission is established. For the nonlinear and non convex problem of air pressure equations, the incremental piecewise linearization method based on the idea of feasible region segmentation is used to transform it into MILP model, and the effective optimal solution is obtained. Finally, an example is given to verify the effectiveness and accuracy of the proposed algorithm.

Comparing solution paths of sparse quadratic minimization with a Stieltjes matrix

This paper studies several solution paths of sparse quadratic minimization problems as a function of the weighing parameter of the bi-objective of estimation loss versus solution sparsity. Three such paths are considered: the “ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document}-path” where the discontinuous ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document}-function provides the exact sparsity count; the “ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path” where the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-function provides a convex surrogate of sparsity count; and the “capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path” where the nonconvex nondifferentiable capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-function aims to enhance the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-approximation. Serving different purposes, each of these three formulations is different from each other, both analytically and computationally. Our results deepen the understanding of (old and new) properties of the associated paths, highlight the pros, cons, and tradeoffs of these sparse optimization models, and provide numerical evidence to support the practical superiority of the capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path. Our study of the capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path is interesting in its own right as the path pertains to computable directionally stationary (= strongly locally minimizing in this context, as opposed to globally optimal) solutions of a parametric nonconvex nondifferentiable optimization problem. Motivated by classical parametric quadratic programming theory and reinforced by modern statistical learning studies, both casting an exponential perspective in fully describing such solution paths, we also aim to address the question of whether some of them can be fully traced in strongly polynomial time in the problem dimensions. A major conclusion of this paper is that a path of directional stationary solutions of the capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-regularized problem offers interesting theoretical properties and practical compromise between the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document}-path and the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path. Indeed, while the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document}-path is computationally prohibitive and greatly handicapped by the repeated solution of mixed-integer nonlinear programs, the quality of ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path, in terms of the two criteria—loss and sparsity—in the estimation objective, is inferior to the capped ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-path; the latter can be obtained efficiently by a combination of a parametric pivoting-like scheme supplemented by an algorithm that takes advantage of the Z-matrix structure of the loss function.

A Gauss–Newton-based decomposition algorithm for Nonlinear Mixed-Integer Optimal Control Problems

For the fast approximate solution of Mixed-Integer Non-Linear Programs (MINLPs) arising in the context of Mixed-Integer Optimal Control Problems (MIOCPs) a decomposition algorithm exists that solves a sequence of three comparatively less hard subproblems to determine an approximate MINLP solution. In this work, we propose a problem formulation for the second algorithm stage that is a convex approximation of the original MINLP and relies on the Gauss–Newton approximation. We analyze the algorithm in terms of approximation properties and establish a first-order consistency result. Then, we investigate the proposed approach considering a numerical case study of Mixed-Integer Optimal Control (MIOC) of a renewable energy system. The investigation shows that the proposed formulation can yield an improved integer solution regarding the objective of the original MINLP compared with the established Combinatorial Integral Approximation (CIA) algorithm.

Simultaneous Synthesis and Design of Reaction–Separation–Recycle Processes Using Rigorous Models

Simultaneous synthesis and design of reaction–separation–recycle processes using rigorous models is highly desirable to improve process efficiency. However, it often leads to a large-scale highly challenging optimization problem. In this work, we propose a computationally efficient optimization framework for the challenging problem. The reactor and separator networks are modeled using the generalized disjunctive programming, which are reformulated into a highly nonconvex mixed-integer nonlinear programming (MINLP) formulation using the convex-hull method. To solve the complex MINLP model, a systematic solution approach is proposed in which an initialization strategy is first proposed to generate a feasible solution for a partially relaxed synthesis problem using the hybrid steady-state and time-relaxation optimization algorithm. A successive relaxed MINLP solution strategy is then adopted to solve the original model to local optimality. The computational results demonstrate that the proposed framework obtains better solutions with less computational effort, by ∼1 order of magnitude, than the existing algorithms.

Capacitated facility location–allocation problem for wastewater treatment in an industrial cluster

We present a location–allocation problem for wastewater treatment in a cluster of processing units. The problem involves installing wastewater treatment plants of appropriate capacities at suitable locations and allocating processing units to these plants. The operational cost of treatment plants is a nonconvex function of wastewater volume and pollutant concentration. This problem is formulated as a non-convex mixed integer nonlinear programming (MINLP) problem. For the general class of this problem, solution strategies based on adapted McCormick envelopes and also piece-wise linear approximation are presented. For a special case of this problem, an exact convexification strategy is proposed and an outer approximation based branch-and-cut approach is presented as an exact solution method. For solving larger instances, we present a matheuristic solution method based on multi-start neighborhood search incorporated into a MINLP solution approach. A real-world case study is presented to illustrate the results of the model along with a multi-objective analysis.

Computer-aided design of ionic liquids for hybrid process schemes

Hybrid process schemes that combine two (or more) units operating at their highest process efficiencies to perform one (or more) process tasks are considered as potentially innovative and sustainable processing options. Additionally, Ionic liquids (ILs), as well as certain organic chemicals, are good candidates for use as solvents in hybrid schemes that can replace energy-intensive processing steps. As successful design of solvent-based hybrid schemes depends on the specific properties of the solvent used, a computer-aided ionic liquid design (CAILD) toolbox was added to an existing tool for computer-aided molecular design for solvent selection-design. Promising IL solvent candidates were first identified through the formulation and solution of mixed-integer nonlinear programming (MINLP) problems for CAILD and were then further evaluated in the process simulation design stage, where the process variables were optimized by means of trade-off and sensitivity analysis. In order to understand and develop model-based hybrid reaction systems, a dynamic model that describes the behavior of the reaction system has been developed. Based on a wide range of collected experimental data, parameters of sub-models, used to calculate the temperature-dependent properties of ILs, are regressed for the purpose of process simulation. Consequently, a hybrid process design method combining CAILD and process design-simulation to identify the optimal IL and its corresponding hybrid process specifications has been proposed. The application of this design method has been illustrated through case studies including the separation of aqueous solutions using an IL-based hybrid distillation scheme and the bio-oxidation of alcohols using a hybrid reaction-separation scheme with continuous product removal.

Improved quadratic cuts for convex mixed-integer nonlinear programs

This paper presents scaled quadratic cuts based on scaling the second-order Taylor expansion terms for the decomposition methods Outer Approximation and Partial Surrogate Cuts for solving convex Mixed Integer Nonlinear Programing problems. The scaled quadratic cut is proved to be a stricter and tighter underestimation for convex nonlinear functions than classical supporting hyperplanes, which results in the improvement of Outer Approximation and Partial Surrogate Cuts based solution methods. We integrate the strategies of scaled quadratic cuts with multi-generation cuts for Outer Approximation and Partial Surrogate Cuts and develop six types of Mixed Integer Nonlinear Programming solution methods with scaled quadratic cuts. These cuts are incorporated in the master problem of the decomposition methods leading to a Mixed Integer Quadratically Constrained Programming problem. Numerical results of benchmark Mixed Integer Nonlinear Programming problems demonstrate the effectiveness of the proposed Mixed Integer Nonlinear Programming solution methods with scaled quadratic cuts.

Outer approximation methods for the solution of co-design optimisation problems in water distribution networks

In the present manuscript, we investigate and demonstrate the use of outer approximation methods for simultaneously optimising the placement and operation of control valves in water distribution networks. The problem definition results in a mixed-integer nonlinear program with nonconvex constraints. We simplify the formulation, compared to previous literature, in order to reduce the degree of nonlinearity in the constraints and decrease the total problem size. We then formulate the application of outer approximation based methods for the generation of good quality local optimal solutions for the considered co-design problem. Finally, we present the results of applying the developed techniques to two case studies, and also comparing the performances of the outer approximation approaches with those of other local mixed integer nonlinear programming solution methods.

Nonlinear Model-Based Process Operation under Uncertainty Using Exact Parametric Programming

Abstract In the present work, two new, (multi-)parametric programming (mp-P)-inspired algorithms for the solution of mixed-integer nonlinear programming (MINLP) problems are developed, with their main focus being on process synthesis problems. The algorithms are developed for the special case in which the nonlinearities arise because of logarithmic terms, with the first one being developed for the deterministic case, and the second for the parametric case (p-MINLP). The key idea is to formulate and solve the square system of the first-order Karush-Kuhn-Tucker (KKT) conditions in an analytical way, by treating the binary variables and/or uncertain parameters as symbolic parameters. To this effect, symbolic manipulation and solution techniques are employed. In order to demonstrate the applicability and validity of the proposed algorithms, two process synthesis case studies are examined. The corresponding solutions are then validated using state-of-the-art numerical MINLP solvers. For p-MINLP, the solution is given by an optimal solution as an explicit function of the uncertain parameters.

Inexact feasibility pump for mixed integer nonlinear programming

The mixed integer nonlinear programming (MINLP) problem as an optimization problem involves both continuous and discrete variables. Moreover, at least one of the functions defining the objective function or the constraints must be nonlinear. Because of its complexity, it is very difficult to obtain the exact optimal solution. Therefore, the heuristic methods for getting a feasible solution of MINLPs are very important in practice. The feasibility pump is one of the famous heuristic methods, which alternates between solving nonlinear programming (NLP) problems and mixed integer linear programming (MILP) relaxed master problems.In this paper, we will extend the feasibility pump to the case where the NLP problems are solved inexactly and propose the convergence of this method under some conditions. Moreover, we present the study of inexactness of the Lagrange multipliers (which are returned negative) of the NLP subproblems.

Mixed Integer Nonlinear Programming Framework for Fixed Path Coordination of Multiple Underwater Vehicles Under Acoustic Communication Constraints

Mixed integer nonlinear programming (MINLP) techniques are increasingly used to address challenging problems in robotics, especially multivehicle motion planning (MVMP). The main contribution of this paper is a discrete time, distributed receding horizon mixed integer nonlinear programming (RH–MINLP) formulation of the underwater multivehicle path coordination problem with constraints on vehicle kinematics, dynamics, collision avoidance, and acoustic communication connectivity, and the application of state-of-the-art MINLP solution techniques. Each vehicle robot starts from a fixed start point and moves toward a goal point along a fixed path, so as to avoid collisions and remain in communication connectivity with other robots. Acoustic communication connectivity constraints account for the attenuation due to signal propagation and delays arising from multipath propagation in noisy communication environments, and specify intervehicle connectivity in terms of a signal-to-noise ratio (SNR) threshold. Scenarios including up to four robots are simulated to demonstrate: 1) the effect of communication connectivity requirements on robot velocity profiles; and 2) the dependence of the solution computation time on the communication connectivity requirement. Typically the optimization improved connectivity at no appreciable cost in journey time (as measured by the arrival time of the last arriving robot). Results also demonstrate the responsive nature of robot trajectories to safety requirements with collision avoidance being achieved at all times despite overlapping and intersecting paths.

Simultaneous model selection and parameter estimation for lithium‐ion batteries: A sequential MINLP solution approach

Equivalent circuit model (ECM) is a practical and commonly used tool not only in state of charge (SOC) estimation but also in state of health (SOH) monitoring for lithium‐ion batteries (LIBs). The functional forms of circuit parameters with respect to SOC in ECM are usually empirical determined, which cannot guarantee to obtain a compact and simple model. A systematical solution framework for simultaneous functional form selection and parameter estimation is proposed. A bi‐objective mixed‐integer nonlinear programming (MINLP) model is first constructed. Two solution approaches, namely the explicit and implicit methods, are then developed to balance model accuracy and model complexity. The former explicitly treats the model complexity as a constraint and the latter implicitly embeds the model complexity into the objective as a penalty. Both approaches require sequential solution of the transformed MINLP model and an ideal and nadir ideal solutions‐based criterion is utilized to terminate the solution procedure for determining the optimal functional forms, in which ideal solution and nadir ideal solution represent the best and worst of each objective, respectively. Both explicit and implicit approaches are thoroughly evaluated and compared through experimental pulse current discharge test and hybrid pulse power characterization test of a commercial LIB. The fitting and prediction results illustrate that the proposed methods can effectively construct an optimal ECM with minimum complexity and prescribed precision requirement. It is thus indicated that the proposed MINLP‐based solution framework, which could automatically guide the optimal ECM construction procedure, can be greatly helpful to both SOC estimation and SOH monitoring for LIBs. © 2015 American Institute of Chemical Engineers AIChE J, 62: 78–89, 2016

A hierarchical method to integrated solvent and process design of physical CO2 absorption using the SAFT‐γ Mie approach

Molecular‐level decisions are increasingly recognized as an integral part of process design. Finding the optimal process performance requires the integrated optimization of process and solvent chemical structure, leading to a challenging mixed‐integer nonlinear programming (MINLP) problem. The formulation of such problems when using a group contribution version of the statistical associating fluid theory, SAFT‐γ Mie, to predict the physical properties of the relevant mixtures reliably over process conditions is presented. To solve the challenging MINLP, a novel hierarchical methodology for integrated process and solvent design (hierarchical optimization) is presented. Reduced models of the process units are developed and used to generate a set of initial guesses for the MINLP solution. The methodology is applied to the design of a physical absorption process to separate carbon dioxide from methane, using a broad selection of ethers as the molecular design space. The solvents with best process performance are found to be poly(oxymethylene)dimethylethers. © 2015 American Institute of Chemical Engineers AIChE J, 61: 3249–3269, 2015

Reactive Power and Voltage Control in Distribution Systems With Limited Switching Operations

An algorithm based on a nonlinear interior-point method and discretization penalties is proposed in this paper for the solution of the mixed-integer nonlinear programming (MINLP) problem associated with reactive power and voltage control in distribution systems to minimize daily energy losses, with time-related constraints being considered. Some of these constraints represent limits on the number of switching operations of transformer load tap changers (LTCs) and capacitors, which are modeled as discrete control variables. The discrete variables are treated here as continuous variables during the solution process, thus transforming the MINLP problem into an NLP problem that can be more efficiently solved exploiting its highly sparse matrix structure; a strategy is developed to round these variables off to their nearest discrete values, so that daily switching operation limits are properly met. The proposed method is compared with respect to other well-known MINLP solution methods, namely, a genetic algorithm and the popular GAMS MINLP solvers BARON and DICOPT. The effectiveness of the proposed method is demonstrated in the well-known PG&E 69-bus distribution network and a real distribution system in the city of Guangzhou, China, where the proposed technique has been in operation since 2003.

Global solution of optimization problems with signomial parts

In this paper a new approach for the global solution of nonconvex MINLP (Mixed Integer NonLinear Programming) problems that contain signomial (generalized geometric) expressions is proposed and illustrated. By applying different variable transformation techniques and a discretization scheme a lower bounding convex MINLP problem can be derived. The convexified MINLP problem can be solved with standard methods. The key element in this approach is that all transformations are applied termwise. In this way all convex parts of the problem are left unaffected by the transformations. The method is illustrated by four example problems.

A kriging based method for the solution of mixed-integer nonlinear programs containing black-box functions

In this paper a new methodology is developed for the solution of mixed-integer nonlinear programs under uncertainty whose problem formulation is complicated by both noisy variables and black-box functions representing a lack of model equations. A branch-and-bound framework is employed to handle the integer complexity whereby the solution to the relaxed nonlinear program subproblem at each node is obtained using both global and local information. Global information is obtained using kriging models which are used to identify promising neighborhoods for local search. Response surface methodology (RSM) is then employed whereby local models are sequentially optimized to refine the problem's lower and upper bounds. This work extends the capabilities of a previously developed kriging-response surface method enabling a wider class of problems to be addressed containing integer decisions and black box models. The proposed algorithm is applied to several small process synthesis examples and its effectiveness is evaluated in terms of the number of function calls required, number of times the global optimum is attained, and computational time.

Mixed-integer nonlinear programming: Some modeling and solution issues

We examine various aspects of modeling and solution via mixed-integer nonlinear programming (MINLP). MINLP has much to offer as a powerful modeling paradigm. Recently, significant advances have been made in MINLP solution software. To fully realize the power of MINLP to solve complex business optimization problems, we need to develop knowledge and expertise concerning MINLP modeling and solution methods. Some of this can be drawn from conventional wisdom of mixed-integer linear programming (MILP) and nonlinear programming (NLP), but theoretical and practical issues exist that are specific to MINLP. This paper discusses some of these, concentrating on an aspect of a classical facility location problem that is well-known in the MILP literature, although here we consider a nonlinear objective function.

Waste heat integration between processes

Energy integration between several processes has been applied giving more economical and profitable operation. The method for waste heat integration of several processes was analysed to reduce energy usage in three possible steps: 1. retrofit of the individual processes, 2. analysis of efficient transfer of waste heat between: ◦ existing non-retrofitted processes ◦ existing non-retrofitted and efficiently retrofitted processes ◦ efficient retrofits and 3. simultaneous integration between some non-retrofitted and some retrofitted existing processes using mixed integer non-linear programming (MINLP). The integration between the processes can be carried out by all the steps, or by the first and the third steps or only by the last step depending on the complexity of the problems. All the steps can be carried out simultaneously and the MINLP algorithm applied with or without the first and the second step. The approach has been illustrated by four complex processes using pinch analysis and MINLP. The optimal solution of pinch analysis and MINLP yields the same additional savings of 526 000 USD/a and selects the same structure and parameters as the MINLP one.

An extended cutting plane method for solving convex MINLP problems

An extended version of Kelley's cutting plane method is introduced in the present paper. The extended method can be applied for the solution of convex MINLP (mixed-integer non-linear programming) problems, while Kelley's cutting plane method was originally introduced for the solution of convex NLP (non-linear programming) problems only. The method is suitable for solving large convex MINLP problems with a moderate degree of nonlinearity. The convergence properties of the method are given in the present paper and an example is provided to illustrate the numerical procedure.

MINLP solution for an optimal isotope separation system

This paper deals with designing of cryogenic distillation systems for the separation of hydrogen isotopes in a thermonuclear fusion process. The design must minimize the tritium inventory in the distillation columns and satisfy the separation requirements. This induces the optimization of both the structure and the operating conditions of the columns. Such a problem is solved by use of a Mixed-Integer NonLinear Programming (MINLP) tool coupled to a process simulator. The MINLP procedure is based on the iterative and alternative treatment of two subproblems : a MLP problem which is solved by a reduced-gradient method, and a MILP problem, solved with a Branch and Boimd method coupled to a simplexe. The formulation of the problem and the choice of an appropriate superstructure are here detailed, and results are finally presented, concerning the optimal design of a specific isotope separation system.