Nonconvex Mixed-integer Nonlinear Programming Research Articles

Optimization of fractionation schemes and beamlet intensities in intensity-modulated radiation therapy with changing cancer tumor properties

Intensity-modulated radiation therapy (IMRT) is a type of external beam radiation therapy used in cancer treatment. In IMRT, the prescribed radiation dose can be administered such that it is maximized on the cancerous tumor while sparing the surrounding healthy tissues. The total dose is divided into fractions across time intervals, called a fractionation scheme. To find the best fractionation scheme and beamlet intensities for total dose, optimization models are used. In this paper, a non-convex mixed-integer nonlinear programming model has been proposed wherein the spatiotemporal changes of the biological properties of the tumor due to tumor cell re-oxygenation, redistribution, and re-population that occur as the treatment progresses have been considered. Also, the dose constraints over both cumulative limits and per-fraction limits have been considered in the model. The output of this model is called the fractionation scheme and beamlet intensities considering biological changes in tumor cells (FBBTs). When the FBBTs are compared with conventional fractionation scheme and beamlet intensities (CFB) which do not include the biological properties of the tumor, it is observed that the FBBTs are more efficacious than the CFBs. To get FBBTs for datasets that resemble realistic tumors, an algorithm based on simulated annealing has been developed and used.

On Enhancing Resilience to Cascading Failures via Post-Disturbance Tweaking of Line Reactances

The primary goal of this paper is to develop an optimization framework for studying the efficacy of transmission line reactance tweaking as a mechanism for post-disturbance control in a transmission network. We start by developing a mixed-integer linear programming (MILP) formulation for tracking the redistribution of direct current (DC) flows and the graph-theoretic evolution of a network topology over the course of cascading failures. Next, we propose a min–max setup for studying the impact of post-disturbance reactance tweaking on the resilience of the system to a worst-case disturbance. We devise a MILP reformulation scheme for the underlying bilevel non-convex mixed-integer nonlinear program to facilitate the computation of its globally optimal solution. We then develop a MILP framework for computing a tight upper bound (best-case scenario) on the efficacy of post-disturbance reactance tweaking for a given set of bus loads. Our numerical case study suggests that post-disturbance reactance tweaking, even on only a small number of lines, can be effective in reducing the amount of load shed in some scenarios in the tested system.

Parallel shuttle bus service design for planned mass rapid transit shutdown: The Singapore experience

19 MRT stations in Singapore experienced a two full-day shutdown as planned in order to upgrade the infrastructure of aging mass rapid transit (MRT) lines. A large bus fleet, running entirely parallel to MRT lines, was deployed to ferry passengers between affected stations, which however results in prolonged congestion and queue of buses near terminals. Rail transit systems in many metropolitan areas are threatened because of aging infrastructure. This study, motivated by the Singapore case, proposes a novel parallel shuttle bus service design (PSBSD) problem that addresses route design, terminal selection, berth allocation, and bus deployment to minimize inconvenience caused to passengers, which is seldom addressed in previous studies. We further develop a mixed-integer nonlinear programming (MINLP) model for the PSBSD problem. In order to solve the non-convex MINLP model optimally, we put forward an effective decomposition method, which comprises (i) the generation of candidate shuttle bus routes through a brute-force search for semi-route and a terminal selection procedure and (ii) the deployment of buses and assignment of terminal berths for selected shuttle bus routes by using an optimization-based approach. Numerical experiments were conducted using data sets from the MRT closure incident in Singapore. The results demonstrate that the decomposition method is capable of finding optimal solutions. Useful insights are also obtained from testing the models under different scenarios.

Joint Uplink/Downlink Sub-Channel, Bit and Time Allocation for Multi-Access Edge Computing

We consider an orthogonal frequency division multiple access (OFDMA)-based Multi-access Edge Computing (MEC) system, consisting of one serving node and multiple users each with an inelastic computation task of a non-negligible task processing duration and a non-negligible computation result size. A joint uplink/downlink sub-channel, bit and time allocation problem is investigated to minimize the energy consumption, which happens to be a very challenging non-convex mixed integer nonlinear programming (MINLP) problem. We equivalently convert it into a convex MINLP problem by using the McCormick envelope, and develop two low-complexity algorithms to obtain two suboptimal solutions. Specifically, one is based on continuous relaxation with greedy rounding and the other one bases on penalty convex–concave procedure. Simulation results show the advantages of our suboptimal solutions.

A Mixed-Integer Convex Programming Algorithm for Security-Constrained Unit Commitment of Power System with 110-kV Network and Pumped-Storage Hydro Units

The secure operation of 110-kV networks should be considered in the optimal generation dispatch of regional power grids in large central cities. However, since 110-kV lines do not satisfy the premise of R << X in the direct current power flow (DCPF) model, the DCPF, which is mostly applied in the security-constrained unit commitment (SCUC) problem of high-voltage power grids, is no longer suitable for describing the active power flow of regional power grids in large central cities. Hence, the quadratic active power flow (QAPF) model considering the resistance of lines is proposed to describe the network security constraints, and an SCUC model for power system with 110-kV network and pumped-storage hydro (PSH) units is established. The analytical expressions of the spinning reserve (SR) capacity of PSH units are given considering different operational modes, and the SR capacity of PSH units is included in the constraint of the SR capacity requirement of the system. The QAPF is a set of quadratic equality constraints, making the SCUC model a mixed-integer nonlinear non-convex programming (MINNP) model. To reduce the computational complexity of solving the model when applied in actual large-scale regional networks, the QAPF model is relaxed by its convex hull, and the SCUC model is transformed into a mixed-integer convex programming (MICP) model, which can be solved to obtain the global optimal solution efficiently and reliably by the mature commercial solver GUROBI (24.3.3, GAMS Development Corporation, Guangzhou, China). Test results on the IEEE-9 bus system, the PEGASE 89 bus system and the Shenzhen city power grid including the 110-kV network demonstrate that the relaxed QAPF model has good calculation accuracy and efficiency, and it is suitable for solving the SCUC problem in large-scale regional networks.

A Mixed-Integer Second-Order Cone Programming Algorithm for the Optimal Power Distribution of AC-DC Parallel Transmission Channels

For the controllability of the transmission power of DC transmission channels, the optimal power distribution (OPD) of AC-DC parallel transmission channels is an effective measure for improving the economic operation of an AC-DC interconnected power grid. A dynamic optimal power flow model for day-ahead OPD of AC-DC parallel transmission channels is established in this paper. The power flow equation constraints of an AC-DC interconnected power grid and the constraints of the discrete regulation requirement of the transmission power of DC channels are considered, which make the OPD model of the AC-DC parallel transmission channels a mixed-integer nonlinear non-convex programming (MINNP) model. Through a cone relaxation transformation and a big M method equivalent transformation, the non-convex terms in the objective function and constraints are executed with the convex relaxation, and the MINNP model is transformed into a mixed-integer second-order cone programming model that can be solved reliably and efficiently using the mature optimization solver GUROBI. Taking an actual large-scale AC-DC interconnected power grid as an example, the results show that the OPD scheme of the AC-DC parallel transmission channels obtained by the proposed algorithm can effectively improve the economical operation of an AC-DC interconnected power grid.

Optimization of thermo-hydraulic systems using multiparametric delay modeling

Thermo-hydraulic networks are playing an important role in current and future energy systems as they allow to use waste heat and cold improving the overall energy efficiency of the energy system. Additionally, they enable an efficient co-generation of e.g. electricity and heat in one process. Thus, optimal operation of thermo-hydraulic systems is of high interest. Finding an optimal solution for this problem is very challenging, as the energy balances of thermo-hydraulic systems are non-convex being influenced by bilinear terms as well as variable dependent time delays for temperature propagation. Hence, if commitment decisions of several heat or cold generation units are considered, the resulting problem is a non-convex mixed integer nonlinear program (MINLP). Past publications mostly used sequential or iterative approaches to solve this problem not approaching a global optimum. Hence, the quality of their solutions cannot be properly evaluated as no guarantees of convergence to global optimality are given and it is unknown if better solutions exist. In this paper we use multiparametric disaggregation for global optimization of bilinear terms and propose “multiparametric delay modeling” for optimization of variable dependent time delays. Combining the two allows to calculate the gap to global optimality and, hence, results can be used to benchmark other optimization approaches for thermo-hydraulic systems.

Stochastic optimal dispatch of PV/wind/diesel/battery microgrids using state‐space approximate dynamic programming

In the operation of microgrids (MGs), the stochastic production of solar/wind, the discrete variables of photovoltaic (PV) inverter's auxiliary service state and diesel generators’ (DGs’) off–on state generally need to be considered, and a stochastic mixed-integer non-linear non-convex programming (MINNP) model is established for the optimal dispatch of MGs. In this model, the expected value of the sum of DGs’ operation as well as start-up cost, the network-loss cost and the PV inverter's auxiliary service cost, is considered as the objective function. The stochastic MINNP model is transformed into a stochastic mixed-integer second-order cone programming (MISOCP) model to reduce the computational complexity. The state-space approximate dynamic programming algorithm is adopted to solve the stochastic MISOCP model. In the algorithm, based on the approximate value functions of typical states that are computed according to the Markov decision process, solving the optimisation model of multiple periods is executed by solving each period's optimisation model one by one to improve the computational efficiency. Meanwhile, parallel computing is executed to greatly improve the efficiency of the proposed algorithm. Test results on two modified IEEE-33 bus and IEEE-123 bus islanded MGs with PV/wind/diesel/battery demonstrate the correctness and efficiency of the proposed model and algorithm.

Faster algorithms for min-max-min robustness for combinatorial problems with budgeted uncertainty

We consider robust combinatorial optimization problems where the decision maker can react to a scenario by choosing from a finite set of k solutions. This approach is appropriate for decision problems under uncertainty where the implementation of decisions requires preparing the ground. We focus on the case that the set of possible scenarios is described through a budgeted uncertainty set and provide three algorithms for the problem. The first algorithm solves heuristically the dualized problem, a non-convex mixed-integer non-linear program (MINLP), via an alternating optimization approach. The second algorithm solves the MINLP exactly for k=2 through a dedicated spatial branch-and-bound algorithm. The third approach enumerates k-tuples, relying on strong bounds to avoid a complete enumeration. We test our methods on shortest path instances that were used in the previous literature and on randomly generated knapsack instances, and find that our methods considerably outperform previous approaches. Many instances that were previously not solved within hours can now be solved within few minutes, often even faster.

A joint decomposition method for global optimization of multiscenario nonconvex mixed-integer nonlinear programs

This paper proposes a joint decomposition method that combines Lagrangian decomposition and generalized Benders decomposition, to efficiently solve multiscenario nonconvex mixed-integer nonlinear programming (MINLP) problems to global optimality, without the need for explicit branch and bound search. In this approach, we view the variables coupling the scenario dependent variables and those causing nonconvexity as complicating variables. We systematically solve the Lagrangian decomposition subproblems and the generalized Benders decomposition subproblems in a unified framework. The method requires the solution of a difficult relaxed master problem, but the problem is only solved when necessary. Enhancements to the method are made to reduce the number of the relaxed master problems to be solved and ease the solution of each relaxed master problem. We consider two scenario-based, two-stage stochastic nonconvex MINLP problems that arise from integrated design and operation of process networks in the case study, and we show that the proposed method can solve the two problems significantly faster than state-of-the-art global optimization solvers.

Carbon-neutral hybrid energy systems with deep water source cooling, biomass heating, and geothermal heat and power

This article addresses the optimal design of carbon-neutral hybrid energy system with deep water source cooling, biomass heating, and geothermal heat and power. A novel superstructure of the proposed hybrid energy system comprised of an enhanced geothermal system, a torrefied biomass-based combustion system, and a deep water source cooling system with conventional chillers as auxiliaries, is developed. Based on the superstructure of the proposed hybrid energy system, we develop a multi-period optimization model to minimize a fractional metric, the levelized cost. Because the main product of the hybrid energy system is heat, the levelized cost is expressed as levelized cost of heat, with other byproducts indirectly incorporated by using credits. The resulting nonconvex mixed-integer nonlinear fractional programming problem is efficiently solved by using a tailored optimization algorithm. Two case studies based on Cornell’s campus in Ithaca, New York are presented to quantify the effect of different electric power sources on the technoeconomic objective, as well as the life cycle greenhouse gas emissions. The first case study considers electric power from natural gas, while a carbon-neutral electric power supply based on renewable geothermal energy is envisioned in the second case study. Results show that switching the electric power supply from natural gas to geothermal energy could reduce the greenhouse gas emissions by 24.5%, while only increasing the levelized cost of heat by 5.6%. The carbon footprint for both case studies are promisingly low, compared with numerous existing heat generation technologies. Through sensitivity analysis, the project lifetime is identified as the most influential input parameter.

Coordinated Operation of Electricity and Natural Gas Systems: A Convex Relaxation Approach

The variability in the generation dispatch of the natural gas generation units will lead to fluctuation in natural gas demand profile that could further jeopardize the security of the natural gas network. The coordinated operation of electricity and natural gas infrastructure systems would help to improve the security and reliability measures in both infrastructure systems and mitigate the risk of demand curtailment. The electricity and natural gas network operation problems are non-convex mixed-integer nonlinear programming problems that are hard to solve in polynomial time. The non-convex feasible regions are formed by the Weymouth constraint and the introduced binary commitment decision variables in the natural gas and electricity network operation problems, respectively. This paper utilized a sparse semidefinite programming (SDP) relaxation to procure the optimal solution for the coordinated operation of electricity and natural gas networks. The presented algorithm leverages the sparseness of the natural gas network to construct several small matrices of lifting variables that are used to form a tight and traceable SDP relaxation. A set of valid constraints that tighten the relaxation ensures the exactness of the solution procured from the relaxed problem. The effectiveness of the presented approach is shown in case studies.

Model Reduction and Outer Approximation for Optimizing the Placement of Control Valves in Complex Water Networks

The optimal placement and operation of pressure control valves in water distribution networks is a challenging engineering problem. When formulated in a mathematical optimization framework, this problem results in a nonconvex mixed integer nonlinear program (MINLP), which has combinatorial computational complexity. As a result, the considered MINLP becomes particularly difficult to solve for large-scale looped operational networks. We extend and combine network model reduction techniques with the proposed optimization framework in order to lower the computational burden and enable the optimal placement and operation of control valves in these complex water distribution networks. An outer approximation algorithm is used to solve the considered MINLPs on reduced hydraulic models. We demonstrate that the restriction of the considered optimization problem on a reduced hydraulic model is not equivalent to solving the original larger MINLP, and its solution is therefore sub-optimal. Consequently, we investigate the trade-off between reducing computational complexity and the potential sub-optimality of the solutions that can be controlled with a parameter of the model reduction routine. The efficacy of the proposed method is evaluated using two large scale water distribution network models.

Robust joint user association and resource partitioning in heterogeneous cloud RANs with dual connectivity

Dual connectivity (DC) allows users to connect to both a macrocell and a small cell simultaneously and is introduced to combat constrained fronthaul links while improving throughput and user mobility. In this paper, we investigate the problem of joint user association and resource allocation in heterogeneous cloud radio access networks (H-CRAN) with DC. The problem is formulated as a non-convex mixed integer non-linear program with constraints on non-ideal fronthaul links and quality of service (QoS) requirements. We propose a block coordinate descent (BCD)-based algorithm that successively solves associated subproblems and prove its convergence. We also utilize successive convex approximation (SCA) to deal with binary association variables. Moreover, we discuss a realistic case in which imperfect knowledge of channel state information causes the user’s rate to be uncertain at the time of decision making. Then, we extend our model to a robust optimization problem by introducing protection functions for constraints. We introduce parameters to address the trade-off between robustness and performance and derive probability bounds of constraint violation. Our numerical results validate the effectiveness of both our nominal and robust algorithms in uncertain environments, confirm the convergence of our proposed algorithms, and examine the cost of robustness.

A risk-averse stochastic program for integrated system design and preventive maintenance planning

The failure of high-consequence systems, such as highspeed railways, can result in a series of severe damages. Due to the volatility of real circumstances, stochastic optimization methods are needed to aid decisions on reliability design and maintenance for the high-consequence systems. Traditionally, risk-neutral approaches are used by considering the expectation of random variables as a preference criterion. The risk-neutral approaches may achieve the solutions that are good in the long run but do not control poor results under certain realizations of random variables. From a perspective of risk analysis, such solutions are not acceptable for high-consequence systems. To address this issue, this paper uses the conditional value at risk (CVaR) to more properly account for some of the worst realizations of random future usage scenarios, and then proposes a risk-averse two-stage stochastic programming model to simultaneously determine the numbers of components in each subsystem and preventive maintenance time intervals for the high-consequence systems that are exposed to uncertain future usage stresses. The proposed stochastic programming model can be converted to a nonconvex mixed-integer nonlinear programming (MINLP) model. To solve the model, we derive the analytical properties of the recourse function and the closed form of CVaR and then design a decomposition algorithm. Numerical examples demonstrate the proposed risk-averse stochastic approach and the effectiveness of incorporating the CVaR in modeling. The results show the research problem significantly benefits from the proposed approach. Furthermore, the robustness of the optimal system design and maintenance plan under different profiles of future usage scenarios is addressed.

Energy-efficient multi-cell resource allocation in cognitive radio-enabled 5G systems

In this paper, we propose an energy-efficient resource allocation (RA) algorithm in cognitive radio-enabled 5th generation (5G) systems, where the scenario including one primary system and multiple secondary cells is considered. Because of the high spectrum leakage of traditional orthogonal frequency division multiplexing (OFDM), alternative modulation schemes regarded as the potential air interfaces in 5G are analyzed, e.g., filter bank-based multi-carrier (FBMC), generalized frequency division multiplexing (GFDM), and universal filtered multi-carrier (UFMC). Our objective is to maximize the whole energy efficiency of secondary system defined by the ratio of the capacity to the total power consumption subject to some practical constraints. The general formulation leads to a non-convex mixed-integer nonlinear programming problem with fractional structure, which is challenging to solve due to its intractability and significant complexity. Therefore, we resort to an alternate optimization framework to optimize the variables of subcarrier assignment and power allocation, where successive convex approximation (SCA) is employed so that the general formulation is finally transformed into a solvable convex problem. Numerical results validate the effectiveness of the proposed RA algorithm, and the comparison with some existing RA algorithms is conducted. In addition, the performance of using different 5G candidate waveforms in the energy-efficient RA algorithm is also presented and discussed.

Price-and-verify: a new algorithm for recursive circle packing using Dantzig–Wolfe decomposition

Packing rings into a minimum number of rectangles is an optimization problem which appears naturally in the logistics operations of the tube industry. It encompasses two major difficulties, namely the positioning of rings in rectangles and the recursive packing of rings into other rings. This problem is known as the Recursive Circle Packing Problem (RCPP). We present the first dedicated method for solving RCPP that provides strong dual bounds based on an exact Dantzig--Wolfe reformulation of a nonconvex mixed-integer nonlinear programming formulation. The key idea of this reformulation is to break symmetry on each recursion level by enumerating one-level packings, i.e., packings of circles into other circles, and by dynamically generating packings of circles into rectangles. We use column generation techniques to design a "price-and-verify" algorithm that solves this reformulation to global optimality. Extensive computational experiments on a large test set show that our method not only computes tight dual bounds, but often produces primal solutions better than those computed by heuristics from the literature.

An adaptive discretization algorithm for the design of water usage and treatment networks

In this paper, we consider the design of water usage and treatment systems in industrial plants. In such a system, the demand of water using units as well as environmental regulations for wastewater have to be met. To this end, water treatment units have to be installed and operated to remove contaminants from the water. The objective of the design problem is to simultaneously optimize the network structure and water allocation of the system at minimum total cost. Due to many bilinear mass balance constraints, this water allocation problem is a nonconvex mixed integer nonlinear program (MINLP) where nonlinear solvers have difficulties to find feasible solutions for real world instances. Therefore, we present a problem specific algorithm to iteratively solve this MINLP. In each iteration, this algorithm deals with an interplay of a mixed integer linear program (MILP) and a quadratically constrained program (QCP). First, an MILP approximates the original problem via discretization and provides a suitable network structure. Then, by fixing this structure, the original MINLP turns into a QCP which yields feasible solutions to the original problem. To improve the accuracy of the generated structure, the discretization of the MILP is adapted after each iteration based on the previous MILP solution. In many cases where nonlinear solvers fail, this approach leads to feasible solutions with good solution quality in short running time.

Packing circles into perimeter-minimizing convex hulls

We present and solve a new computational geometry optimization problem in which a set of circles with given radii is to be arranged in unspecified area such that the length of the boundary, i.e., the perimeter, of the convex hull enclosing the non-overlapping circles is minimized. The convex hull boundary is established by line segments and circular arcs. To tackle the problem, we derive a non-convex mixed-integer non-linear programming formulation for this circle arrangement or packing problem. Moreover, we present some theoretical insights presenting a relaxed objective function for circles with equal radius leading to the same circle arrangement as for the original objective function. If we minimize only the sum of lengths of the line segments, for selected cases of up to 10 circles we obtain gaps smaller than $$10^{-4}$$ using BARON or LINDO embedded in GAMS, while for up to 75 circles we are able to approximate the optimal solution with a gap of at most $$14\%$$ .

Global optimality bounds for the placement of control valves in water supply networks

This manuscript investigates the problem of optimal placement of control valves in water supply networks, where the objective is to minimize average zone pressure. The problem formulation results in a nonconvex mixed integer nonlinear program (MINLP). Due to its complex mathematical structure, previous literature has solved this nonconvex MINLP using heuristics or local optimization methods, which do not provide guarantees on the global optimality of the computed valve configurations. In our approach, we implement a branch and bound method to obtain certified bounds on the optimality gap of the solutions. The algorithm relies on the solution of mixed integer linear programs, whose formulations include linear relaxations of the nonconvex hydraulic constraints. We investigate the implementation and performance of different linear relaxation schemes. In addition, a tailored domain reduction procedure is implemented to tighten the relaxations. The developed methods are evaluated using two benchmark water supply networks and an operational water supply network from the UK. The proposed approaches are shown to outperform state-of-the-art global optimization solvers for the considered benchmark water supply networks. The branch and bound algorithm converges to good quality feasible solutions in most instances, with bounds on the optimality gap that are comparable to the level of parameter uncertainty usually experienced in water supply network models.