Nonconvex MINLP Research Articles

On the use of overlapping convex hull relaxations to solve nonconvex MINLPs

AbstractWe present a novel relaxation for general nonconvex sparse MINLP problems, called overlapping convex hull relaxation (CHR). It is defined by replacing all nonlinear constraint sets by their convex hulls. If the convex hulls are disjunctive, e.g. if the MINLP is block-separable, the CHR is equivalent to the convex hull relaxation obtained by (standard) column generation (CG). The CHR can be used for computing an initial lower bound in the root node of a branch-and-bound algorithm, or for computing a start vector for a local-search-based MINLP heuristic. We describe a dynamic block and column generation (DBCG) MINLP algorithm to generate the CHR by dynamically adding aggregated blocks. The idea of adding aggregated blocks in the CHR is similar to the well-known cutting plane approach. Numerical experiments on nonconvex MINLP instances show that the duality gap can be significantly reduced with the results of CHRs. DBCG is implemented as part of the CG-MINLP framework Decogo, see https://decogo.readthedocs.io/en/latest/index.html.

A feasible path-based branch and bound algorithm for strongly nonconvex MINLP problems

In this paper, a feasible path-based branch and bound (B&B) algorithm is proposed to solve mixed-integer nonlinear programming problems with highly nonconvex nature through integration of the previously proposed hybrid feasible-path optimisation algorithm and the branch and bound method. The main advantage of this novel algorithm is that our previously proposed hybrid steady-state and time-relaxation-based optimisation algorithm is employed to solve a nonlinear programming (NLP) subproblem at each node during B&B. The solution from a parent node in B&B is used to initialize the NLP subproblems at the child nodes to improve computational efficiency. This approach allows circumventing complex initialisation procedure and overcoming difficulties in convergence of process simulation. The capability of the proposed algorithm is illustrated by several process synthesis and intensification problems using rigorous models.

Wireless power transfer supported device-to-device multicast energy cooperative transmission scheme

The cluster heads (CHs) will face the contradiction between the lack of available energy and the demand of high energy consumption in energy harvesting-supported device-to-device multicast communications (EH-D2MD). Wireless power transfer (WPT) technology is a possible way to address the above contradiction. The members of a device-to-device (D2D) multicast cluster can transfer part of their available energy to the CH by WPT and jointly undertake contents unloading. As a result, the transmission robustness of the multicast cluster can be improved. Therefore, a transmission scheme with energy cooperation (EC) is designed on the premise of cellular spectrum reusing. The EC scheme designs elaborately the energy harvesting, energy cooperation and data transmission of the spectrum reusing process. To realize the EC scheme, this work is to maximize the transmission rate of a multicast cluster and give the joint optimal solution of multi-domain resources including spectrum resource allocation, cooperative time factor planning, and power control. The rate maximization problem is a typical non-convex mixed integer non-linear programming (non-convex MINLP) problem. To investigate the performance of EH-D2MD communication scenario, a convex approximate lower-bound algorithm is proposed, which can transfer the non-convex problem into convex MINLP and can give a joint solution. Simulation results show that the proposed algorithm obtains a lower-bound solution of the rate maximization problem in comparison with the exhausted searching method. Furthermore, compared with the scheme without energy cooperation, the established EC transmission scheme can increase the transmission rate of D2MD by more than 45% and enhance the robustness of EH-D2MD.

Simulation-based optimization of distillation processes using an extended cutting plane algorithm

The use of commercial flowsheeting programs enables straight-forward use of rigorous, but user hidden, mathematical formulations of chemical processes. The optimization of such black-box models is a challenging task due to nonconvexity, absence of accurate derivatives, and simulation convergence failures which can prevent classical optimization procedures from continuing the search. Here, we present an optimization framework based on the extended cutting plane algorithm with additional heuristic techniques and strategies designed to improve its practical performance for solving nonconvex simulation-based MINLP problems. The new algorithmic features include two approaches for dealing with nonconvexities; the first technique expands the search space to restore feasibility of the MILP subproblems, and the second is a restarting technique to avoid premature termination to non-optimal solutions. We also propose two approaches for handle simulation failures, based on no-good cuts and backtracking. The proposed optimization framework is successfully applied to four case studies dealing with the economic optimization of distillation processes.

A column generation algorithm for solving energy system planning problems

Energy system optimization models are typically large models which combine sub-models which range from linear to very nonlinear. Column generation (CG) is a classical tool to generate feasible solutions of sub-models, defining columns of global master problems, which are used to steer the search for a global solution. In this paper, we present a new inner approximation method for solving energy system MINLP models. The approach is based on combining CG and the Frank Wolfe algorithm for generating an inner approximation of a convex relaxation and a primal heuristic for computing solution candidates. The features of this approach are: (i) no global branch-and-bound tree is used, (ii) sub-problems can be solved in parallel to generate columns, which do not have to be optimal, nor become available at the same time to synchronize the solution, (iii) an arbitrary solver can be used to solve sub-models, (iv) the approach (and the implementation) is generic and can be used to solve other nonconvex MINLP models. We perform experiments with decentralized energy supply system models with more than 3000 variables. The numerical results show that the new decomposition method is able to compute high-quality solutions and has the potential to outperform state-of-the-art MINLP solvers.

Efficient Two-Phase Algorithm to Solve Nonconvex MINLP Model of Pump Scheduling Problem

AbstractIn water distribution networks, pumps are used to raise the pressure of water and transfer it throughout the network. Because the cost of electricity consumed by pumps is very high, optimal...

Multiperiod optimization of heat exchanger networks with integrated thermodynamic cycles and thermal storages

This work proposes a multiperiod synthesis methodology to optimize simultaneously the utility systems, Rankine cycles and Heat Exchanger Networks considering different expected operating modes and off-design operating conditions. Heat exchangers are modeled with different approaches depending on the type of off-design control measure. The problem is formulated as a nonconvex MINLP. Being too challenging for general purpose MINLP solvers, a bilevel decomposition algorithm is specifically developed. One case study consists in designing a flexible Organic Rankine Cycle able to deal with different operating modes, and the returned solution shows a considerable improvement in economics compared to a single-period design. The other two case studies are an Integrated Gasification Combined Cycle able to operate in two different modes, and a flexible Integrated Solar Combined Cycle power plant. Despite the large number of hot/cold streams and technical design/operational constraints, the proposed method can find very good solutions featuring cost-effective designs.

Integrated optimization of design, storage sizing, and maintenance policy as a Markov decision process considering varying failure rates

Various strategies can be applied to improve reliability at certain costs, including equipment redundancy, product storage, and maintenance, which gives rise to the problem of optimally allocating the reliability improvement costs among the strategies and balancing them against the potential loss due to unavailabilities. Motivated by the reliability concerns of air separation units, we use Markov Decision Process to model the stochastic dynamic decision making process of condition-based maintenance assuming bathtub shaped failure rate curves of single units, which is then embedded into a non-convex MINLP (DMP) that considers the trade-off among all the decisions. An initial attempt to directly solve the MINLP (DMP) for a mid-sized problem with several global solvers reveals severe computational difficulties. In response, we propose a custom two-phase algorithm that greatly reduces the required computation effort. The algorithm also shows consistent performance over randomly generated problems around the original example of 4 processing stages and problems of larger sizes.

Global optimization algorithm for multi-period design and planning of centralized and distributed manufacturing networks

This paper addresses the design and planning of manufacturing networks considering the option of centralized and distributed facilities, taking into account the potential trade-offs between investments and transportation. The problem is formulated as an extension of the Capacitated Multi-facility Weber Problem, which involves the selection of which facilities to build in each time period, and their location in the continuous two-dimensional space, in order to meet demand and minimize costs. The model is a multi-period GDP, reformulated as a nonconvex MINLP. We propose an accelerated version of the Bilevel Decomposition by Lara et al. (2018) that finds stronger bounds in the decomposition scheme. We benchmark the performance of our algorithm against the original Bilevel Decomposition and commercial global solvers and show that our approach outperforms the others in all instances tested. Additionally, we illustrate the applicability of the proposed model and solution framework with a biomass supply chain case study.

Holistic Planning Model for Sustainable Water Management in the Shale Gas Industry

To address water planning decisions in shale gas operations, we present a novel water management optimization model that explicitly takes into account the effect of high concentrations of total dissolved solids (TDS) and temporal variations in the impaired water. The model comprises different water management strategies: (a) direct wastewater reuse, which is possible because of new additives tolerant to high TDS concentrations but at the expense of increasing the costs; (b) wastewater treatment, separately taking into account pretreatment, softening, and desalination technologies; and (c) the use of Class II disposal sites. The objective is to maximize the “sustainability profit” by determining the flowback destination (reuse, degree of treatment, or disposal), the fracturing schedule, the fracturing-fluid composition, and the number of water-storage tanks needed for each period of time. Because of the rigorous determination of TDS in all water streams, the model is a nonconvex MINLP model that is tackled...

Including fluctuations of water content in feed streams and products for the optimal management of water resources

In the food industry, water is frequently used both as a system utility and as a component that enters products and sub products. In other words, fresh water entering the process exits it either in the wastewaters or in the products. Consequently, the usual optimisation techniques for minimizing the overall water consumption are to be modified accordingly. In particular, the presence of chemical reactions in the balance equations can give rise to severe nonlinearities which are to be taken into account in the preliminary reconciliation of the process measurements, as well as in the following optimisation step. Furthermore, the presence of bilinear terms (flow-rates multiplied by concentrations) results in the model being non-convex. The objective function employed is the overall NPV function that considers both capital and operation costs, using a conventional 5 % discount factor. If all treatment options for the wastewaters and all possible interactions among the units are considered for the overall optimisation, the resulting superstructure gives rise to a stochastic non-convex MINLP problem, which was solved by combining an order optimisation approach with the Baron algorithm in the GAMS environment, as well as with software available in the public domain. No substantial difference in accuracy and efficiency between the two algorithms was observed. The method is applied to a complex process for the manufacture of starches and starch products and analysed the reduction in fresh water consumption as a result of the optimisation procedure.

On a nonconvex MINLP formulation of the Euclidean Steiner tree problem in n-space: missing proofs

We supply proofs for a few key results concerning smoothing square roots and model strengthening for a mixed-integer nonlinear-optimization formulation of the the Euclidean Steiner tree problem.

Lossless convexification of nonconvex MINLP on the UAV path‐planning problem

SummaryMost unmanned aerial vehicle path‐planning problems have been modeled as linear optimal control problems, ie, as mixed‐integer linear programming problems. However, most constraints cannot be described accurately in linear form in practical engineering applications. In this paper, the traditional unmanned aerial vehicle path‐planning problem is modified as a nonconvex mixed‐integer nonlinear programming problem, whose continuous relaxation is a nonconvex programming problem. A lossless convexification method is introduced into the generalized Benders decomposition algorithm framework. Thus, an optimal solution can be obtained without directly solving the nonconvex programming problem. The output of the proposed algorithm has been rigorously proved to be the optimal solution to the original problem. Meanwhile, the simulation results verify the validity of the theoretical analysis and demonstrate the superior efficiency of the proposed algorithm.

Mixed-integer nonlinear programming models for optimal design of reliable chemical plants

Motivated by reliability/availability concerns in chemical plants, this paper proposes MINLP models to determine the optimal selection of parallel units considering the trade-off between availability and cost. Assuming an underlying serial structure for availability, we consider first a case where the system transitions between available and unavailable states, and second the case with an intermediate state at half capacity. Two non-convex MINLP models maximizing net profit are introduced for the two cases. In addition, a bi-criterion MINLP model is proposed to maximize availability and to minimize cost for the first case. It is shown that the corresponding epsilon-constrained model, where the availability is maximized subject to parametrically varying upper bound of the cost, can be reformulated as a convex MINLP. Availability is also incorporated in the superstructure optimization of process flowsheets. The performances of the proposed models are illustrated with a methanol synthesis and a toluene hydrodealkylation process.

Orbital shrinking: Theory and applications

We present a method, based on formulation symmetry, for generating Mixed-Integer Linear Programming (MILP) relaxations with fewer variables than the original symmetric MILP. Our technique also extends to convex MINLP, and some nonconvex MINLP with a special structure. We showcase the effectiveness of our relaxation when embedded in a decomposition method applied to two important applications (multi-activity shift scheduling and multiple knapsack problem), showing that it can improve CPU times by several orders of magnitude compared to pure MIP or CP approaches.

MINLP model and two-stage algorithm for the simultaneous synthesis of heat exchanger networks, utility systems and heat recovery cycles

This work proposes a novel approach for the simultaneous synthesis of Heat Exchanger Networks (HEN) and Utility Systems of chemical processes and energy systems. Given a set of hot and cold process streams and a set of available utility systems, the method determines the optimal selection, arrangement and design of utility systems and the heat exchanger network aiming to rigorously consider the trade-off between efficiency and capital costs. The mathematical formulation uses the SYNHEAT superstructure for the HEN, and ad hoc superstructures and nonlinear models to represent the utility systems. The challenging nonconvex MINLP is solved with a two-stage algorithm. A sequential synthesis algorithm is specifically developed to generate a good starting solution. The algorithm is tested on a literature test problem and two industrial problems, the optimization of the Heat Recovery Steam Cycle of a Natural Gas Combined Cycle and the heat recovery system of an Integrated Gasification Combined Cycle.

Three ideas for a feasibility pump for nonconvex MINLP

We describe an implementation of the Feasibility Pump heuristic for nonconvex MINLPs. Our implementation takes advantage of three novel techniques, which we discuss here: a hierarchy of procedures for obtaining an integer solution, a generalized definition of the distance function that takes into account the nonlinear character of the problem, and the insertion of linearization cuts for nonconvex constraints at every iteration. We implemented this new variant of the Feasibility Pump as part of the global optimization solver Couenne. We present experimental results that compare the impact of the three discussed features on the ability of the Feasibility Pump to find feasible solutions and on the solution quality.

An MILP-MINLP decomposition method for the global optimization of a source based model of the multiperiod blending problem

The multiperiod blending problem involves binary variables and bilinear terms, yielding a nonconvex MINLP. In this work we present two major contributions for the global solution of the problem. The first one is an alternative formulation of the problem. This formulation makes use of redundant constraints that improve the MILP relaxation of the MINLP. The second contribution is an algorithm that decomposes the MINLP model into two levels. The first level, or master problem, is an MILP relaxation of the original MINLP. The second level, or subproblem, is a smaller MINLP in which some of the binary variables of the original problem are fixed. The results show that the new formulation can be solved faster than alternative models, and that the decomposition method can solve the problems faster than state of the art general purpose solvers.

Mathematical Programming Methods for Pressure Management in Water Distribution Systems

In this paper, we survey mathematical programming methods for the management of pressure in water distribution systems through optimal placement and operation of control valves. The optimization framework addresses the minimization of average zone pressure under multiple demand scenarios, enforcing hydraulic equations as nonlinear constraints. Binary variables are used to model the placement of valves. The derived nonlinear optimization problem is a non-convex MINLP. We implement and evaluate a direct MINLP solver and two different reformulation methods for MINLPs that solve sequences of regular NLPs. Moreover, we investigate the solutions under different design and operation loading conditions.

Extended cutting plane method for a class of nonsmooth nonconvex MINLP problems

In this article, a generalization of the ECP algorithm to cover a class of nondifferentiable Mixed-Integer NonLinear Programming problems is studied. In the generalization constraint functions are required to be -pseudoconvex instead of pseudoconvex functions. This enables the functions to be nonsmooth. The objective function is first assumed to be linear but also -pseudoconvex case is considered. Furthermore, the gradients used in the ECP algorithm are replaced by the subgradients of Clarke subdifferential. With some additional assumptions, the resulting algorithm shall be proven to converge to a global minimum.