Nonlinear Programming Subproblems Research Articles

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.

Multidisciplinary Design Optimization Approach to Integrated Space Mission Planning and Spacecraft Design

Space mission planning and spacecraft design are tightly coupled and need to be considered together for optimal performance; however, this integrated optimization problem results in a large-scale mixed-integer nonlinear programming (MINLP) problem, which is challenging to solve. In response to this challenge, this paper proposes a new solution approach to this problem based on decomposition-based optimization via augmented Lagrangian coordination. The proposed approach leverages the unique structure of the problem that enables its decomposition into a set of coupled subproblems of different types: a mixed-integer quadratic programming (MIQP) subproblem for mission planning, and one or more nonlinear programming (NLP) subproblem(s) for spacecraft design. Because specialized MIQP or NLP solvers can be applied to each subproblem, the proposed approach can efficiently solve the otherwise intractable integrated MINLP problem. An automatic and effective method to find an initial solution for this iterative approach is also proposed so that the optimization can be performed without a user-defined initial guess. The demonstration case study shows that, compared to the state-of-the-art method, the proposed formulation converges substantially faster and the converged solution is at least the same or better given the same computational time limit.

A Mathematical Negotiation Mechanism for Distributed Procurement Problems and a Hybrid Algorithm for its Solution

In this paper, a mathematical negotiation mechanism is designed to minimize the negotiators’ costs in a distributed procurement problem at two echelons of an automotive supply chain. The buyer’s costs are procurement cost and shortage penalty in a one-period contract. On the other hand, the suppliers intend to solve a multi-period, multi-product production planning to minimize their costs. Such a mechanism provides an alignment among suppliers’ production planning and order allocation, also supports the partnership with the valued suppliers by taking suppliers’ capacities into account. Such a circumstance has been modeled via bi-level programming, in which the buyer acts as a leader, and the suppliers individually appear as followers in the lower level. To solve this nonlinear bi-level programming model, a hybrid algorithm by combining the particle swarm optimization (PSO) algorithm with a heuristic algorithm based on A* search is proposed. The heuristic A* algorithm is embedded to solve the mixed-integer nonlinear programming (MINLP) sub-problems for each supplier according to the received variable values determined by PSO system particles (buyer’s request for quotations (RFQs)). The computational analyses have shown that the proposed hybrid algorithm called PSO-A* outperforms PSO-SA and PSO-Greedy algorithms.

A Branch and Bound Algorithm for Transmission Network Expansion Planning Using Nonconvex Mixed-Integer Nonlinear Programming Models

The branch and bound (BB) algorithm is widely used to obtain the global solution of mixed-integer linear programming (MILP) problems. On the other hand, when the traditional BB structure is directly used to solve nonconvex mixed-integer nonlinear programming (MINLP) problems, it becomes ineffective, mainly due to the nonlinearity and nonconvexity of the feasible region of the problem. This article presents the difficulties and ineffectiveness of the direct use of the traditional BB algorithm for solving nonconvex MINLP problems and proposes the formulation of an efficient BB algorithm for solving this category of problems. The algorithm is formulated taking into account particular aspects of nonconvex MINLP problems, including (i) how to deal with the nonlinear programming (NLP) subproblems, (ii) how to detect the infeasibility of an NLP subproblem, (iii) how to treat the nonconvexity of the problem, and (iv) how to define the fathoming rules. The proposed BB algorithm is used to solve the transmission network expansion planning (TNEP) problem, a classical problem in power systems optimization, and its performance is compared with the performances of off-the-shelf optimization solvers for MINLP problems. The results obtained for four test systems, with different degrees of complexity, indicate that the proposed BB algorithm is effective for solving the TNEP problem with and without considering losses, showing equal or better performance than off-the-shelf optimization solvers.

Outer Approximation Method for the Unit Commitment Problem with Wind Curtailment and Pollutant Emission

This paper considers the fast and effective solving method for the unit commitment (UC) problem with wind curtailment and pollutant emission in power systems. Firstly, a suitable mixed-integer quadratic programming (MIQP) model of the corresponding UC problem is presented by some linearization techniques, which is difficult to solve directly. Then, the MIQP model is solved by the outer approximation method (OAM), which decomposes the MIQP into a mixed-integer linear programming (MILP) master problem and a nonlinear programming (NLP) subproblem for alternate iterative solving. Finally, simulation results for six systems with up to 100 thermal units and one wind unit in 24 periods are presented, which show the practicality of MIQP model and the effectiveness of OAM.

An expected-cost realization-probability optimization approach for the dynamic energy management of microgrid

Dynamic energy management (DEM) is being recognized as an important tool for the economical and efficient operation of microgrid. This paper proposes an expected-cost realization-probability (ECRP) optimization approach for the DEM of microgrid. In the model, the expected cost is taken as an economic goal to find the schedule with the maximum ECRP (i.e., the maximum probability of obtaining an actual operation cost which is lower than this economic goal). It is a stochastic mixed-integer nonlinear programming (MINLP) problem composed of the AC power flow, battery operation characters and logical variables. To solve this NP-hard problem, an ECRP-based approximate dynamic programming is proposed. The original MINLP problem is decomposed into small nonlinear programming subproblems to facilitate the solution. And the ECRP-based value function is formed to describe the relationship among the decisions of subproblems, which helps to obtain the near-optimal decision in a decomposed manner. Simulation examples presented on a real microgrid indicate that the ECRP of the proposed method is higher than those of the other two comparison algorithms.

An improved algorithm for synthesis of heat exchanger network with a large number of uncertain parameters

This paper presents an improved method for the Mixed Integer Nonlinear Programming (MINLP) synthesis of flexible Heat Exchanger Network with a large number of uncertain parameters. Typically, such a problem is written as a multi-scenario two-stage stochastic model with recourse which is difficult to solve because the size of the model grows exponentially with the number of uncertain parameters. The exponential growth could be avoided by decomposing the model into simpler problems that are solved sequentially in a small number of scenarios.In this work, the determination of the first-stage variables (process topology and unit sizes) and the second-stage variables (operating and control variables) is decomposed into several steps where they are determined separately. This is achieved by iteratively solving smaller two-scenario MINLP and one-scenario Nonlinear Programming (NLP) subproblems. In this way, the size of the problem remains independent of the number of uncertain parameters.The innovation of this study is the introduction of correction factors for the second-stage variables into the objective function when determining the first-stage variables. In this way, better trade-offs are found that mitigate the unpleasant effect of decomposition. The synthesis of the heat exchanger network shows that the implementation of correction factors improves the optimal result by 7.6%.

Robust optimal control for a batch nonlinear enzyme-catalytic switched time-delayed process with noisy output measurements

In this paper, we consider a nonlinear switched time-delayed (NSTD) system with an unknown time-varying function describing the batch culture. The output measurements are noisy. According to the actual fermentation process, this time-varying function appears in the form of a piecewise-linear function with unknown kinetic parameters and switching times. The quantitative definition of biological robustness is given to overcome the difficulty of accurately measuring intracellular material concentrations. Our main goal is to estimate these unknown quantities by using noisy output measurements and biological robustness. This estimation problem is formulated as a robust optimal control problem (ROCP) governed by the NSTD system subject to continuous state inequality constraints. The ROCP is approximated as a sequence of nonlinear programming subproblems by using some techniques. Due to the highly complex nature of these subproblems, we propose a hybrid parallel algorithm, based on Nelder–Mead method, simulated annealing and the gradients of the constraint functions, for solving these subproblems. The paper concludes with simulation results.

Partially distributed outer approximation

This paper presents a novel partially distributed outer approximation algorithm, named PaDOA, for solving a class of structured mixed integer convex programming problems to global optimality. The proposed scheme uses an iterative outer approximation method for coupled mixed integer optimization problems with separable convex objective functions, affine coupling constraints, and compact domain. PaDOA proceeds by alternating between solving large-scale structured mixed-integer linear programming problems and partially decoupled mixed-integer nonlinear programming subproblems that comprise much fewer integer variables. We establish conditions under which PaDOA converges to global minimizers after a finite number of iterations and verify these properties with an application to thermostatically controlled loads and to mixed-integer regression.

Optimized Layout of Spherical Objects in a Polyhedral Domain

Introduction. The article studies the problem of arranging spherical objects in a bounded polyhedral domain in order to maximize the packing factor. The spherical objects have variable placement parameters and variable radii within the given upper and lower bounds. The constraints on the allowable distance between each pair of spherical objects are taken into account. The phi-function technique is used for analytical description of the placement constraints, involving object non-overlapping and containment conditions. The problem is considered as a nonlinear programming problem. The feasible region is described by a system of inequalities with differentiable functions. To find the local maximum of the problem the decomposition algorithm is used. We employ the strategy of active set of inequalities for reducing the computational complexity of the algorithm. IPOPT solver for solving nonlinear programming subproblems is used. The multistart strategy allows selecting the best local maximum point. Numerical results and the appropriate graphic illustration are given. The purpose of the article is presenting a mathematical model and developing a solution algorithm for arranging spherical objects in a polyhedral region with the maximum packing factor. It allows obtaining a locally optimal solution in a reasonable time. Results. A new formulation of the problem of arranging spherical objects in a polyhedral domain is considered, where both the placement parameters and the radii of the spherical objects are variable. A mathematical model in the form of nonlinear programming problem is derived. A solution approach based on the decomposition algorithm and multistart strategy is developed. The numerical results combined with the graphical illustration are given. Conclusions. The proposed approach allows modeling optimized layouts of spherical objects into a polyhedral domain. Keywords: layout, spherical objects, polyhedral domain, phi-function.

Sample average approximation for stochastic nonconvex mixed integer nonlinear programming via outer-approximation

We propose a sample average approximation-based outer-approximation algorithm (SAAOA) that can address nonconvex two-stage stochastic programs (SP) with any continuous or discrete probability distributions. Previous work has considered this approach for convex two-stage SP (Wei and Realff in Comput Chem Eng 28(3):333–346, 2004). The SAAOA algorithm does internal sampling within a nonconvex outer-approximation algorithm where we iterate between a mixed-integer linear programming (MILP) master problem and a nonconvex nonlinear programming (NLP) subproblem. We prove that the optimal solutions and optimal value obtained by the SAAOA algorithm converge to the optimal solutions and the optimal value of the true SP problem as the sample size goes to infinity. The convergence rate is also given to estimate the sample size. Since the theoretical sample size estimate is too conservative in practice, we propose an SAAOA algorithm with confidence intervals for the upper bound and the lower bound at each iteration of the SAAOA algorithm. Two policies are proposed to update the sample sizes dynamically within the SAAOA algorithm with confidence intervals. The proposed algorithm works well for the special case of pure binary first stage variables and continuous stage two variables since in this case the nonconvex NLPs can be solved for each scenario independently. The proposed algorithm is tested with a stochastic pooling problem and is shown to outperform the external sampling approach where large scale MINLPs need to be solved.

Optimal minimal variation control with quality constraint for fed-batch fermentation processes involving multiple feeds

In this paper, an optimal minimal variation control in the fed-batch fermentation of glycerol bioconversion to 1,3-propanediol (1,3-PD) caused by Klebsiella pneumonia (K. pneumonia) is considered. In fed-batch process, it is required for the concentration of 1,3-PD to satisfy a quality constraint (i.e., the concentration of 1,3-PD at the terminal time is to reach a critical value), and furthermore, the cost associated with the change of the control signal must also be taken into account. It is formulated as an optimal control problem in which the total variation of the control signal is taken as the cost function, while the feeding rates of glycerol and alkali and the switching instants are taken as decision variables. The optimal control problem is subject to the quality constraint and certain continuous state inequality constraints. To handle the continuous state inequality constraints, the optimal control problem is approximated as a sequence of nonlinear programming subproblems. Because of the nature of the optimal control problem, which is highly complicated, a parallel algorithm is proposed to solve these subproblems based on genetic algorithm and the gradients of the constraint functions with respect to decision variables. From extensive simulation studies, it is observed that the obtained optimal feeding rates and switching instants are highly satisfactory.

Robust optimal control problem with multiple characteristic time points in the objective for a batch nonlinear time-varying process using parallel global optimization

In this paper, we consider a nonlinear time-varying dynamical (NTVD) system with uncertain system parameters assigned to their nominal values in batch culture of glycerol bioconversion to 1,3-propanediol induced by Klebsiella pneumoniae. Some important properties of the NTVD system are discussed. Our goal is to choose a time-varying function for the NTVD system. Thus, an optimal control problem (OCP) governed by the NTVD system and subject to continuous state inequality constraints arising from engineering specifications is proposed, where the time-varying function is the control function to be chosen such that system cost (the relative error between experimental data and the simulated output of the system) and system robustness (robustness of the system with respect to uncertain system parameters) is optimized. Based on the actual fermentation process, the time-varying function is specified by a four-piecewise linear function with unknown kinetic parameters and switching instants. The resulting OCP is approximated as a sequence of nonlinear mathematical programming subproblems by the time-scaling transformation, the constraint transcription and the locally smoothing approximation techniques. A parallel global optimization algorithm, based on a novel combination of limited information particle swarm optimization and local search strategy, is then developed to solve these subproblems. Numerical results show the effectiveness and applicability of our proposed algorithm.

Optimal layout of ellipses and its application for additive manufacturing

The paper studies a layout problem of variable number of ellipses with variable sizes placed into an arbitrary disconnected polygonal domain with maximum packing factor. The ellipses can be continuously translated and rotated. Restrictions on the dimensions of the ellipses are taken into account. Tools for the mathematical modelling of placement constraints (distance constraints between ellipses and containment of ellipses into a polygonal domain) using the phi-function technique are introduced. The tools make it possible to formulate the layout problem in the form of MIP model that is equivalent to a sequence of nonlinear programming subproblems. We develop a new solution algorithm that involves the feasible starting point algorithm and optimisation procedure to search for efficient locally optimal solutions of the layout problem. This algorithm can be used in the design of parts for «support-free» additive manufacturing, taking into account the conditions for its static/ dynamic strength. Results of the algorithm implementation for a topologically optimised flat part with the analysis of a stress state are provided.

Decomposition Algorithm for Optimization Placement Problems

The paper considers a placement problem of 2D convex objects in a rectangular domain of minimum area, that related to the field of Packing and Cutting problems. Our objects may be continuously translated and rotated. A nonlinear programming model of the problem is derived using the phi-function technique. We develop an efficient decomposition algorithm to search for local optimal solutions for the placement problem. The algorithm reduces our problem to a sequence of nonlinear programming subproblems of considerably smaller dimension and a smaller number of nonlinear inequalities. The benefit of this approach is borne out by the computational results

A reduced regularization strategy for economic NMPC

Nonlinear Model Predictive Control (NMPC) enables the incorporation of detailed dynamic process models for nonlinear, multivariable control with constraints. This optimization-based framework also leads to on-line dynamic optimization with performance-based and so-called economic objectives. Nevertheless, economic NMPC (eNMPC) still requires careful formulation of the nonlinear programming (NLP) subproblem to guarantee stability. In this study, we derive a novel reduced regularization eNMPC approach with stability guarantees. Compared with full state regularization, the proposed strategy is less conservative and easier to implement. The resulting eNMPC framework is firstly demonstrated on a nonlinear continuous stirred-tank reactor (CSTR) example and a large-scale double distillation system example. Then the proposed strategy is applied to a challenging nonlinear CO2 capture model, where bubbling fluidized bed models comprise a solid-sorbent post-combustion carbon capture system. Our results indicate the benefits of this improved eNMPC approach over tracking to the setpoint, and better stability over eNMPC without regularization.

Two-step optimization procedure for the conceptual design of A-frame systems for solar power plants

This work presents a two-stage optimization procedure for the conceptual design and operation of A-frame dry cooling systems for concentrated solar power facilities. First, the optimal geometry of the A-frame including sizing, number of fans and blade geometry, and unit parameters such as pipe length, configuration and number is determined. Finally, the operation of the system over a year for minimum energy consumption is computed. The geometry problem is formulated as a mixed-integer non linear programming (MINLP) problem. A tailor-made branch and bound algorithm is used to solve the complex non-linear programming sub-problems. The second problem consists of a multi-period MINLP. A fixed geometry is used to evaluate the usage of fans over time. The solution suggests an apex angle of 63°, one row of 75 pipes of 13.5 m long with a diameter of 3.3 mm, and 4 fans are used but they only operate at full capacity during summer. This design allows reducing the energy required by 20% by using the appropriate pipe configuration and number. The unit consumes around 4% of the energy produced by the CSP plant that serves. It is a promising result that can be affected by plant layout and ground availability.

Approximate Dantzig–Wolfe decomposition to solve a class of variational inequality problems with an illustrative application to electricity market models

In this study, we develop a new Approximate Dantzig–Wolfe (ADW) decomposition method for variational inequalities (VI) based on the work by Chung and Fuller (2010) and Çelebi and Fuller (2013). The decomposed VI consists of one approximate subproblem, which is nonlinear programming (NLP) or linear programming (LP) and one approximate master problem, which is an NLP. Note that we can have many approximate subproblems depending upon the approximation method and the structure of the constraint set. On the other hand, if the VI mapping in the approximate master problem is equal to that in the iterative methods for solving VI, then the ADW-VI simply consists of the computational sequence of solving NLP (or LP) subproblem(s) and NLP master problem in an iterative manner. That is, the iterative methods for VI and the DW decomposition method are combined into a single iterative loop. The details of the method are presented as well as an extension of the theory from Chung and Fuller (2010) and Çelebi and Fuller (2013). In addition, numerical results are provided based on two time-of-use pricing models of Ontario electricity market in Çelebi and Fuller (2013), but for which the new master problem approximation different from Çelebi and Fuller (2013) has been used. These results validate ADW-VI, and in some computational instances, indicates dramatic improvements in solution times as compared to reference methods, like diagonalization method of Dafermos (1983). Another set of numerical results, based on a simple electricity market, illustrates that ADW-VI can be faster than the PATH solver when solving large-scale problem instances.

Extended ADMMs for RPO of large‐scale power systems with discrete controls

The reactive power optimization (RPO) problem of power systems is a mixed-integer nonlinear programming problem, which can be computationally expensive. This paper applies the alternating direction method of multipliers (ADMM) to solve it. By duplicating discrete control variables with one copy allowed to vary continuously, the RPO model is formulated as a modified model in which the objective function and constraints have a separable structure with respect to the continuous and discrete variables, except for the coupling constraint. We then applied the ADMM to solve this model with the separable structure, and hence the original problem is converted into two optimization sub-problems, which are nonlinear programming (NLP) and mixed-integer quadratic programming (MIQP) problems, respectively. To improve the convergence of the algorithm, we proposed an extended ADMM by adding the upper and lower limits on state variables into the MIQP sub-problem based on the sensitivities obtained in the NLP sub-problem. We also established a mechanism to filter out inactive inequality constraints in the MIQP sub-problem to improve computational speed. Moreover, numerical results tested on IEEE test systems and a real 739-bus system have shown the correctness of the proposed method and its adaptability for power systems with different sizes and configurations..

Packing of concave polyhedra with continuous rotations using nonlinear optimisation

We study the problem of packing a given collection of arbitrary, in general concave, polyhedra into a cuboid of minimal volume. Continuous rotations and translations of polyhedra are allowed. In addition, minimal allowable distances between polyhedra are taken into account. We derive an exact mathematical model using adjusted radical free quasi phi-functions for concave polyhedra to describe non-overlapping and distance constraints. The model is a nonlinear programming formulation. We develop an efficient solution algorithm, which employs a fast starting point algorithm and a new compaction procedure. The procedure reduces our problem to a sequence of nonlinear programming subproblems of considerably smaller dimension and a smaller number of nonlinear inequalities. The benefit of this approach is borne out by the computational results, which include a comparison with previously published instances and new instances.