Convex Mixed-integer Nonlinear Programming Research Articles

Computing Optimality Certificates for Convex Mixed-Integer Nonlinear Problems

Every optimization problem has a corresponding verification problem that checks whether a given optimal solution is in fact optimal. In the literature, there are a lot of such ways to verify optimality for a given solution, for example, the branch-and-bound tree. To simplify this task, optimality certificates were introduced for convex mixed-integer nonlinear programs, and it was shown that the sizes of the certificates are bounded in terms of the number of integer variables. We introduce an algorithm to compute the certificates and conduct computational experiments. Through the experiments, we show that the optimality certificates can be surprisingly small. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: This work was supported by the Deutsche Forschungsgemeinschaft [CRC 154 Subproject A05, CRC 154 Subproject B07, and SFB Transregio 154], the Bundesministerium für Wirtschaft und Energie. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0099 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0099 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Comparing perspective reformulations for piecewise-convex optimization

We aim at generalizing formulations for non-convex piecewise-linear problems to mathematical programs whose non-convexities are only expressed in terms of piecewise-convex univariate functions. This is motivated by solving Mixed-Integer Non-Linear Programming (MINLP) problems with separable non-convex functions via the Sequential Convex MINLP technique. We theoretically and computationally compare different formulations, showing that, unlike in the linear case, they are not equivalent when perspective reformulation is applied to strengthen the formulation of each single segment.

Algorithms for a risk-averse Stackelberg game with multiple adversaries

We consider a Stackelberg game that arises in a security domain (SSG), where a defender can simultaneously protect m out of n targets from an adversary that observes the defense strategy before deciding on an utility maximizing attack. Given the high stakes in security settings, it is reasonable that the defender in this game is risk averse with respect to the attacker’s decisions.Here we focus on developing efficient solution algorithms for a specific SSG, where the defender uses an entropic risk measure to model risk aversion to the attacker’s strategies, and where multiple attackers select targets following logit quantal response equilibrium models. This problem can be formulated as a nonconvex nonlinear optimization problem. We propose two solution methods: (1) approximate the problem through convex mixed integer nonlinear programs (MINR) and (2) a general purpose methodology (CELL) to optimize nonconvex and nonseparable fractional problems through mixed integer linear programming approximations. Both methods provide arbitrarily good incumbents and lower bounds on SSG. We present cutting plane methods to solve these problems for large instances. Our computational experiments illustrate the advantages of introducing risk aversion into the defender’s behavior and show that MINR dominates CELL, producing in 2 h solutions that are within 2% of optimal on average.

A Joint Cluster Formation Scheme With Multilayer Awareness for Energy-Harvesting Supported D2D Multicast Communication

Recently, Energy Harvesting-supported Device-to-Device (EH-D2D) communication receives extensive concerns due to its excellent properties in Energy Efficiency (EE), offloading capability, etc. Researches on EH-D2D mainly focus on resource allocation schemes of available energy. With the popularity of mobile devices and applications, content sharing becomes anytime, anywhere. Multicast is the foundation to realize the content sharing and can reduce transmission consumption because of multi-user service feature. Thus, Mutlicast can further improve EE of EH-D2D in communication mode rather than in resource allocation. However, the heterogeneity and imbalance of available energy, content requests and social ties on the user side will bring great challenges to the deployment of EH-D2D Multicast (EH-D2MD) communication. This paper aims at EE optimization and establishes a two-layer cluster formation model, which solves grouping, cluster head selection and power control jointly. When facing the non-convex of the sum of fractional functions and the product of binary variables, this paper specifically designs a two-step Convex Approximation Algorithm (CDA). CDA skillfully transforms the modeled non-convex Mixed Integer Non-Linear Programming (MINLP) problem into a convex MINLP one, which is easier to solve. Simulation results reveal that CDA can obtain the joint cluster formation results with approximate optimal EE and lower complexity.

The supporting hyperplane optimization toolkit for convex MINLP

In this paper, an open-source solver for mixed-integer nonlinear programming (MINLP) problems is presented. The Supporting Hyperplane Optimization Toolkit (SHOT) combines a dual strategy based on polyhedral outer approximations (POA) with primal heuristics. The POA is achieved by expressing the nonlinear feasible set of the MINLP problem with linearizations obtained with the extended supporting hyperplane (ESH) and extended cutting plane (ECP) algorithms. The dual strategy can be tightly integrated with the mixed-integer programming (MIP) subsolver in a so-called single-tree manner, i.e., only a single MIP optimization problem is solved, where the polyhedral linearizations are added as lazy constraints through callbacks in the MIP solver. This enables the MIP solver to reuse the branching tree in each iteration, in contrast to most other POA-based methods. SHOT is available as a COIN-OR open-source project, and it utilizes a flexible task-based structure making it easy to extend and modify. It is currently available in GAMS, and can be utilized in AMPL, Pyomo and JuMP as well through its ASL interface. The main functionality and solution strategies implemented in SHOT are described in this paper, and their impact on the performance are illustrated through numerical benchmarks on 406 convex MINLP problems from the MINLPLib problem library. Many of the features introduced in SHOT can be utilized in other POA-based solvers as well. To show the overall effectiveness of SHOT, it is also compared to other state-of-the-art solvers on the same benchmark set.

Energy-Efficient Parallel Multi-Access Edge Computing in OFDMA Wireless Networks

We study a parallel multi-access edge computing (MEC) system in this work and investigate its energy-efficient resources management. Particularly, the MEC system consists of an edge node and a number of mobile users, where each user has an inelastic computing task with nonnegligible duration of task execution and nonnegligible size of task result. In addition, the MEC system operates in orthogonal frequency division multiple access (OFDMA) wireless networks and can support the transmission as well as the execution of different tasks in parallel. On this basis, we build a nonconvex mixed-integer nonlinear programming (MINLP) problem aiming to minimize the energy consumed by the system. By using the McCormick method and exploring the problem's structural properties, we first equivalently transform it to a manageable convex MINLP problem, and then devise two low-complexity suboptimal algorithms. Finally, numerical results are presented to show the effectiveness of our proposed algorithms.

Using projected cutting planes in the extended cutting plane method

In this paper we show that simple projections can improve the algorithmic performance of cutting plane-based optimization methods. Projected cutting planes can, for example, be used as alternatives to standard cutting planes or supporting hyperplanes in the extended cutting plane (ECP) method. In the paper we analyse the properties of such an algorithm and prove that it will converge to a global optimum for smooth and nonsmooth convex mixed integer nonlinear programming problems. Additionally, we show that we are able to solve two old but very difficult facility layout problems (FLP), with previously unknown optimal solutions, to verified global optimum by using projected cutting planes in the algorithm. These solution results are also given in the paper.

Polyhedral approximation strategies for nonconvex mixed-integer nonlinear programming in SHOT

Different versions of polyhedral outer approximation are used by many algorithms for mixed-integer nonlinear programming (MINLP). While it has been demonstrated that such methods work well for convex MINLP, extending them to solve nonconvex problems has traditionally been challenging. The Supporting Hyperplane Optimization Toolkit (SHOT) is a solver based on polyhedral approximations of the nonlinear feasible set of MINLP problems. SHOT is an open source COIN-OR project, and is currently one of the most efficient global solvers for convex MINLP. In this paper, we discuss some extensions to SHOT that significantly extend its applicability to nonconvex problems. The functionality include utilizing convexity detection for selecting the nonlinearities to linearize, lifting reformulations for special classes of functions, feasibility relaxations for infeasible subproblems and adding objective cuts to force the search for better feasible solutions. This functionality is not unique to SHOT, but can be implemented in other similar methods as well. In addition to discussing the new nonconvex functionality of SHOT, an extensive benchmark of deterministic solvers for nonconvex MINLP is performed that provides a snapshot of the current state of nonconvex MINLP.

A New Combinatorial Algorithm for Separable Convex Resource Allocation with Nested Bound Constraints

The separable convex resource allocation problem with nested bound constraints aims to allocate B units of resources to n activities to minimize a separable convex cost function, with lower and upper bounds on the total amount of resources that can be consumed by nested subsets of activities. We develop a new combinatorial algorithm to solve this model exactly. Our algorithm is capable of solving instances with millions of activities in several minutes. The running time of our algorithm is at most 73% of the running time of the current best algorithm for benchmark instances with three classes of convex objectives. The efficiency of our algorithm derives from a combination of constraint relaxation and divide and conquer based on infeasibility information. In particular, nested bound constraints are relaxed first; if the solution obtained violates some bound constraints, we show that the problem can be divided into two subproblems of the same structure and smaller sizes according to the bound constraint with the largest violation. Summary of Contribution. The resource allocation problem is a collection of optimization models with a wide range of applications in production planning, logistics, portfolio management, telecommunications, statistical surveys, and machine learning. This paper studies the resource allocation model with prescribed lower and upper bounds on the total amount of resources consumed by nested subsets of activities. These nested bound constraints are motivated by storage limits, time-window requirements, and budget constraints in various applications. The model also appears as a subproblem in models for green logistics and machine learning, and it has to be solved repeatedly. The model belongs to the class of computationally challenging convex mixed-integer nonlinear programs. We develop a combinatorial algorithm to solve this model exactly. Our algorithm is faster than the algorithm that currently has the best theoretical complexity in the literature on an extensive set of test instances. The efficiency of our algorithm derives from the combination of an infeasibility-guided divide-and-conquer framework and a scaling-based greedy subroutine for resource allocation with submodular constraints. This paper also showcases the prevalent mismatch between the theoretical worst-case time complexity of an algorithm and its practical efficiency. We have offered some explanations of this mismatch through the perspectives of worst-case analysis, specially designed instances, and statistical metrics of numerical experiments. The implementation of our algorithm is available on an online repository.

Domain Partition of the Hydro Production Function for Solving Efficiently the Short-Term Generation Scheduling Problem

Short-term generation scheduling (STGS) is a fundamental task in the operational planning analysis of hydroelectric plants. For the multi-unit case, the STGS is represented as a large-scale nonconvex mixed-integer nonlinear optimization model. Then, considering the (usual) short time for providing a solution, it is vital to exploit all the structural properties of the problem at hand. The main issue for exploiting this problem is the hydro production function (HPF), which is a nonlinear nonconvex relationship between power, head, and turbined outflow of a generating unit (GU). Nevertheless, the HPF usually presents operating regions where the function is convex and regions in which it is concave due to physical reasons. Inspired by sequential convex mixed-integer nonlinear programming techniques, this paper proposes partitioning the domain of the HPF in regions in which it is convex and those in which it is concave. The HPF is approximated by a piecewise linear function using the logarithmic aggregation convex combination (LACC) model in convex regions. In turn, in the concave regions, the HPF is replaced by a convex hull approach, which, combined with symmetry strategies, reduces the number of binary variables in the resulting optimization problem. Using several computational instances of a 50-unit hydroelectric plant, we show that the partitioning-based proposed strategy significantly reduces the computational time compared to two other efficient MILP formulations.

A proximal-point outer approximation algorithm

Many engineering and scientific applications, e.g. resource allocation, control of hybrid systems, scheduling, etc., require the solution of mixed-integer non-linear problems (MINLPs). Problems of such class combine the high computational burden arising from considering discrete variables with the complexity of non-linear functions. As a consequence, the development of algorithms able to efficiently solve medium-large MINLPs is still an interesting field of research. In the last decades, several approaches to tackle MINLPs have been developed. Some of such approaches, usually defined as exact methods, aim at finding a globally optimal solution for a given MINLP at expense of a long execution time, while others, generally defined as heuristics, aim at discovering suboptimal feasible solutions in the shortest time possible. Among the various proposed paradigms, outer approximation (OA) and feasibility pump (FP), respectively as exact method and as heuristic, deserve a special mention for the number of relevant publications and successful implementations related to them. In this paper we present a new exact method for convex mixed-integer non-linear programming called proximal outer approximation (POA). POA blends the fundamental ideas behind FP into the general OA scheme that attepts to yield faster and more robust convergence with respect to OA while retaining the good performances in terms of fast generation of feasible solutions of FP.

A disjunctive cut strengthening technique for convex MINLP

Generating polyhedral outer approximations and solving mixed-integer linear relaxations remains one of the main approaches for solving convex mixed-integer nonlinear programming (MINLP) problems. There are several algorithms based on this concept, and the efficiency is greatly affected by the tightness of the outer approximation. In this paper, we present a new framework for strengthening cutting planes of nonlinear convex constraints, to obtain tighter outer approximations. The strengthened cuts can give a tighter continuous relaxation and an overall tighter representation of the nonlinear constraints. The cuts are strengthened by analyzing disjunctive structures in the MINLP problem, and we present two types of strengthened cuts. The first type of cut is obtained by reducing the right-hand side value of the original cut, such that it forms the tightest generally valid inequality for a chosen disjunction. The second type of cut effectively uses individual right-hand side values for each term of the disjunction. We prove that both types of cuts are valid and that the second type of cut can dominate both the first type and the original cut. We use the cut strengthening in conjunction with the extended supporting hyperplane algorithm, and numerical results show that the strengthening can significantly reduce both the number of iterations and the time needed to solve convex MINLP problems.

Improving the Representativeness of a Simple Random Sample: An Optimization Model and Its Application to the Continuous Sample of Working Lives

This paper proposes an optimization model for selecting a larger subsample that improves the representativeness of a simple random sample previously obtained from a population larger than the population of interest. The problem formulation involves convex mixed-integer nonlinear programming (convex MINLP) and is, therefore, NP-hard. However, the solution is found by maximizing the size of the subsample taken from a stratified random sample with proportional allocation and restricting it to a p-value large enough to achieve a good fit to the population of interest using Pearson’s chi-square goodness-of-fit test. The paper also applies the model to the Continuous Sample of Working Lives (CSWL), which is a set of anonymized microdata containing information on individuals from Spanish Social Security records and the results prove that it is possible to obtain a larger subsample from the CSWL that (far) better represents the pensioner population for each of the waves analyzed.

Two linear approximation algorithms for convex mixed integer nonlinear programming

We present two new algorithms for convex Mixed Integer Nonlinear Programming (MINLP), both based on the well known Extended Cutting Plane (ECP) algorithm proposed by Weterlund and Petersson. Our first algorithm, Refined Extended Cutting Plane (RECP), incorporates additional cuts to the MILP relaxation of the original problem, obtained by solving linear relaxations of NLP problems considered in the Outer Approximation algorithm. Our second algorithm, Linear Programming based Branch-and-Bound (LP-BB), applies the strategy of generating cuts that is used in RECP, to the linear approximation scheme used by the LP/NLP based Branch-and-Bound algorithm. Our computational results show that RECP and LP-BB are highly competitive with the most popular MINLP algorithms from the literature, while keeping the nice and desirable characteristic of ECP, of being a first-order method.

Joint Optimal Software Caching, Computation Offloading and Communications Resource Allocation for Mobile Edge Computing

As software may be used by multiple users, caching popular software at the wireless edge has been considered to save computation and communications resources for mobile edge computing (MEC). However, fetching uncached software from the core network and multicasting popular software to users have so far been ignored. Thus, existing design is incomplete and less practical. In this paper, we propose a joint caching, computation and communications mechanism which involves software fetching, caching and multicasting, as well as task input data uploading, task executing (with non-negligible time duration) and computation result downloading, and mathematically characterize it. Then, we optimize the joint caching, offloading and time allocation policy to minimize the weighted sum energy consumption subject to the caching and deadline constraints. The problem is a challenging two-timescale mixed integer nonlinear programming (MINLP) problem, and is NP-hard in general. We convert it into an equivalent convex MINLP problem by using some appropriate transformations and propose two low-complexity algorithms to obtain suboptimal solutions of the original non-convex MINLP problem. Specifically, the first suboptimal solution is obtained by solving a relaxed convex problem using the consensus alternating direction method of multipliers (ADMM), and then rounding its optimal solution properly. The second suboptimal solution is proposed by obtaining a stationary point of an equivalent difference of convex (DC) problem using the penalty convex-concave procedure (Penalty-CCP) and ADMM. Finally, by numerical results, we show that the proposed solutions outperform existing schemes and reveal their advantages in efficiently utilizing storage, computation and communications resources.

On the relation between the extended supporting hyperplane algorithm and Kelley\u2019s cutting plane algorithm

Recently, Kronqvist et al. (J Global Optim 64(2):249–272, 2016) rediscovered the supporting hyperplane algorithm of Veinott (Oper Res 15(1):147–152, 1967) and demonstrated its computational benefits for solving convex mixed integer nonlinear programs. In this paper we derive the algorithm from a geometric point of view. This enables us to show that the supporting hyperplane algorithm is equivalent to Kelley’s cutting plane algorithm (J Soc Ind Appl Math 8(4):703–712, 1960) applied to a particular reformulation of the problem. As a result, we extend the applicability of the supporting hyperplane algorithm to convex problems represented by a class of general, not necessarily convex nor differentiable, functions.

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

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

Nonlinear chance-constrained problems with applications to hydro scheduling

We present a Branch-and-Cut algorithm for a class of nonlinear chance-constrained mathematical optimization problems with a finite number of scenarios. Unsatisfied scenarios can enter a recovery mode. This class corresponds to problems that can be reformulated as deterministic convex mixed-integer nonlinear programming problems with indicator variables and continuous scenario variables, but the size of the reformulation is large and quickly becomes impractical as the number of scenarios grows. The Branch-and-Cut algorithm is based on an implicit Benders decomposition scheme, where we generate cutting planes as outer approximation cuts from the projection of the feasible region on suitable subspaces. The size of the master problem in our scheme is much smaller than the deterministic reformulation of the chance-constrained problem. We apply the Branch-and-Cut algorithm to the mid-term hydro scheduling problem, for which we propose a chance-constrained formulation. A computational study using data from ten hydroplants in Greece shows that the proposed methodology solves instances faster than applying a general-purpose solver for convex mixed-integer nonlinear programming problems to the deterministic reformulation, and scales much better with the number of 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.

Improving the performance of DICOPT in convex MINLP problems using a feasibility pump

ABSTRACTThe solver DICOPT is based on the outer-approximation algorithm used for solving mixed-integer nonlinear programming (MINLP) problems. This algorithm is very effective for solving some types of convex MINLPs. However, it has been observed that DICOPT has difficulties solving instances in which some of the nonlinear constraints are so restrictive that nonlinear subproblems generated by the algorithm are infeasible. This problem is addressed in this paper with a feasibility pump algorithm, which modifies the objective function in order to efficiently find feasible solutions. It has been implemented as a preprocessing algorithm, which is used to initialize both the incumbent and the mixed-integer linear relaxation of the outer-approximation algorithm. Computational comparisons with previous versions of DICOPT on a set of convex MINLPs demonstrate the effectiveness of the proposed algorithm in terms of solution quality and solution time.