Comparison of MINLP formulations for global superstructure optimization

Superstructure optimization is a powerful but computationally demanding task that can be used to select the optimal structure among many alternatives within a single optimization. In chemical engineering, such problems naturally arise in process design, where different process alternatives need to be considered simultaneously to minimize a specific objective function (e.g., production costs or global warming impact). Conventionally, superstructure optimization problems are either formulated with the Big-M or the Convex Hull reformulation approach. However, for problems containing nonconvex functions, it is not clear whether these yield the most computationally efficient formulations. We therefore compare the conventional problem formulations with less common ones (using equilibrium constraints, step functions, or multiplications of binary and continuous variables to model disjunctions) using three case studies. First, a minimalist superstructure optimization problem is used to derive conjectures about their computational performance. These conjectures are then further investigated by two more complex literature benchmarks. Our analysis shows that the less common approaches tend to result in a smaller problem size, while keeping relaxations comparably tight—despite the introduction of additional nonconvexities. For the considered case studies, we demonstrate that all reformulation approaches can further benefit from eliminating optimization variables by a reduced-space formulation. For superstructure optimization problems containing nonconvex functions, we therefore encourage to also consider problem formulations that introduce additional nonconvexities but reduce the number of optimization variables.

An interior-point trust-region algorithm to solve a nonlinear bilevel programming problem

<abstract><p>In this paper, a nonlinear bilevel programming (NBLP) problem is transformed into an equivalent smooth single objective nonlinear programming (SONP) problem utilized slack variable with a Karush-Kuhn-Tucker (KKT) condition. To solve the equivalent smooth SONP problem effectively, an interior-point Newton's method with Das scaling matrix is used. This method is locally method and to guarantee convergence from any starting point, a trust-region strategy is used. The proposed algorithm is proved to be stable and capable of generating approximal optimal solution to the nonlinear bilevel programming problem.</p> <p>A global convergence theory of the proposed algorithm is introduced and applications to mathematical programs with equilibrium constraints are given to clarify the effectiveness of the proposed approach.</p></abstract>

Capturing Electricity Market Dynamics in the Optimal Trading of Strategic Agents using Neural Network Constrained Optimization

In competitive electricity markets the optimal trading problem of an electricity market agent is commonly formulated as a bi-level program, and solved as mathematical program with equilibrium constraints (MPEC). In this paper, an alternative paradigm, labeled as mathematical program with neural network constraint (MPNNC), is developed to incorporate complex market dynamics in the optimal bidding strategy. This method uses input-convex neural networks (ICNNs) to represent the mapping between the upper-level (agent) decisions and the lower-level (market) outcomes, i.e., to replace the lower-level problem by a neural network. In a comparative analysis, the optimal bidding problem of a load agent is formulated via the proposed MPNNC and via the classical bi-level programming method, and compared against each other.

A Proximal Algorithm with Convergence Guarantee for a Nonconvex Minimization Problem Based on Reproducing Kernel Hilbert Space

The underlying function in reproducing kernel Hilbert space (RKHS) may be degraded by outliers or deviations, resulting in a symmetry ill-posed problem. This paper proposes a nonconvex minimization model with ℓ0-quasi norm based on RKHS to depict this degraded problem. The underlying function in RKHS can be represented by the linear combination of reproducing kernels and their coefficients. Thus, we turn to estimate the related coefficients in the nonconvex minimization problem. An efficient algorithm is designed to solve the given nonconvex problem by the mathematical program with equilibrium constraints (MPEC) and proximal-based strategy. We theoretically prove that the sequences generated by the designed algorithm converge to the nonconvex problem’s local optimal solutions. Numerical experiment also demonstrates the effectiveness of the proposed method.

Optimal Portfolio Selection Methodology for a Demand Response Aggregator

This paper presents a methodology for determining the optimal portfolio allocation for a demand response aggregator. The formulation is based on Day-Ahead electricity prices, in which the aggregator coordinates a set of residential consumers that are recruited through contracts. Four types of contracts are analyzed, considering both direct and indirect demand response programs. The objective is to compare different scenarios for contract portfolios in order to establish the benefits of each market agent. An optimization problem is formulated to capture the interactions between the aggregator and end consumers. The model is formulated as a mathematical program with equilibrium constraints: At the upper level, the aggregator maximizes its benefits, whereas the lower level represents the consumers’ contracts. By applying the developed methodology, the characterization of the consumers’ behavior is established in order to forecast their responses to the generation of punctual incentives, both for usual scenarios and peak events, as well as to evaluate the impact that direct and indirect control contracts have on the performance of the aggregator as the energy price varies.

A Review on the Complementarity Modelling in Competitive Electricity Markets

In recent years, the ever-increasing research interest in various aspects of the electricity pool-based markets has generated a plethora of complementarity-based approaches to determine participating agents’ optimal offering/bidding strategies and model players’ interactions. In particular, the integration of multiple and diversified market agents, such as conventional generation companies, renewable energy sources, electricity storage facilities and agents with a mixed generation portfolio has instigated significant competition, as each player attempts to establish their market dominance and realize substantial financial benefits. The employment of complementarity modelling approaches can also prove beneficial for the optimal coordination of the electricity and natural gas market coupling. Linear and nonlinear programming as well as complementarity modelling, mainly in the form of mathematical programs with equilibrium constraints (MPECs), equilibrium programs with equilibrium constraints (EPECs) and conjectural variations models (CV) have been widely employed to provide effective market clearing mechanisms, enhance agents’ decision-making process and allow them to exert market power, under perfect and imperfect competition and various market settlements. This work first introduces the theoretical concepts that regulate the majority of contemporary competitive electricity markets. It then presents a comprehensive review of recent advances related to complementarity-based modelling methodologies and their implementation in current competitive electricity pool-based markets applications.

Optimization of Multimodal Discrete Network Design Problems Based on Super Networks

In this paper, we investigate the multimodal discrete network design problem that simultaneously optimizes the car, bus, and rail transit network, in which inter-modal transfers are achieved by slow traffic modes including walking and bike-sharing. Specifically, a super network topology is presented to signify the modal interactions. Then, the generalized cost formulas of each type of links in the super network are defined. And based on the above formulas a bi-objective programming model is proposed to minimize the network operation cost and construction cost with traffic flow equilibrium constraints, investment constraints and expansion constraints. Moreover, a hybrid heuristic algorithm that combines the minimum cost flow algorithm and simulated annealing algorithm is presented to solve the proposed model. Finally, the effectiveness of the proposed model and algorithm is evaluated through two numerical tests: a simple test network and an actual multimodal transport network.