A bilevel optimal motion planning (BOMP) model with application to autonomous parking

In this paper, we present a bilevel optimal motion planning (BOMP) model for autonomous parking. The BOMP model treats motion planning as an optimal control problem, in which the upper level is designed for vehicle nonlinear dynamics, and the lower level is for geometry collision-free constraints. The significant feature of the BOMP model is that the lower level is a linear programming problem that serves as a constraint for the upper-level problem. That is, an optimal control problem contains an embedded optimization problem as constraints. Traditional optimal control methods cannot solve the BOMP problem directly. Therefore, the modified approximate Karush-Kuhn-Tucker theory is applied to generate a general nonlinear optimal control problem. Then the pseudospectral optimal control method solves the converted problem. Particularly, the lower level is the $J_2$-function that acts as a distance function between convex polyhedron objects. Polyhedrons can approximate vehicles in higher precision than spheres or ellipsoids. Besides, the modified $J_2$-function (MJ) and the active-points based modified $J_2$-function (APMJ) are proposed to reduce the variables number and time complexity. As a result, an iteirative two-stage BOMP algorithm for autonomous parking concerning dynamical feasibility and collision-free property is proposed. The MJ function is used in the initial stage to find an initial collision-free approximate optimal trajectory and the active points, then the APMJ function in the final stage finds out the optimal trajectory. Simulation results and experiment on Turtlebot3 validate the BOMP model, and demonstrate that the computation speed increases almost two orders of magnitude compared with the area criterion based collision avoidance method

Packing ellipsoids with overlap

The problem of packing ellipsoids of different sizes and shapes into an ellipsoidal container so as to minimize a measure of overlap between ellipsoids is considered. A bilevel optimization formulation is given, together with an algorithm for the general case and a simpler algorithm for the special case in which all ellipsoids are in fact spheres. Convergence results are proved and computational experience is described and illustrated. The motivating application - chromosome organization in the human cell nucleus - is discussed briefly, and some illustrative results are presented

Progressively interactive evolutionary multiobjective optimization

A complete optimization procedure for a multi-objective problem essentially comprises of search and decision making. Depending upon how the search and decision making task is integrated, algorithms can be classified into various categories. Following `a decision making after search' approach, which is common with evolutionary multi-objective optimization algorithms, requires to produce all the possible alternatives before a decision can be taken. This, with the intricacies involved in producing the entire Pareto-front, is not a wise approach for high objective problems. Rather, for such kind of problems, the most preferred point on the front should be the target. In this study we propose and evaluate algorithms where search and decision making tasks work in tandem and the most preferred solution is the outcome. For the two tasks to work simultaneously, an interaction of the decision maker with the algorithm is necessary, therefore, preference information from the decision maker is accepted periodically by the algorithm and progress towards the most preferred point is made. Two different progressively interactive procedures have been suggested in the dissertation which can be integrated with any existing evolutionary multi-objective optimization algorithm to improve its effectiveness in handling high objective problems by making it capable to accept preference information at the intermediate steps of the algorithm. A number of high objective un-constrained as well as constrained problems have been successfully solved using the procedures. One of the less explored and difficult domains, i.e., bilevel multiobjective optimization has also been targeted and a solution methodology has been proposed. Initially, the bilevel multi-objective optimization problem has been solved by developing a hybrid bilevel evolutionary multi-objective optimization algorithm. Thereafter, the progressively interactive procedure has been incorporated in the algorithm leading to an increased accuracy and savings in computational cost. The efficacy of using a progressively interactive approach for solving difficult multi-objective problems has, therefore, further been justifie

An efficient hybrid differential evolutionary algorithm for zbilevel optimisation problems

Bilevel problems are widely used to describe the decision problems with hierarchical upper–lower-level structures in many economic fields. The bilevel optimisation problem (BLOP) is intrinsically NP-hard when its objectives and constraints are complex and the decision variables are large in scale at both levels. An efficient hybrid differential evolutionary algorithm for BLOP (HDEAB) is proposed where the optimal lower level value function mapping method, the differential evolutionary algorithm, k-near- est neighbours (KNN) and a nested local search are hybridised to improve the computational accuracy and efficiency. To show the performance of the HDEAB, numerical studies were conducted on SMD (Sinha, Maro and Deb) instances and an application example of optimising a venture capital staged-financing contract. The results demonstrate that the HDEAB outperforms the BLEAQ (bile- vel evolutionary algorithm based on quadratic approximations) greatly in solving the BLOPs with different scale

Optimal configuration of energy storage system capacity in traction power supply system considering photovoltaic consumption

In order to achieve energy savings and promote on-site integration of photovoltaic energy in electrified railways, a topology structure is proposed for the integration of photovoltaic (PV) and the energy storage system (ESS) into the traction power supply system (TPSS) based on a railway power conditioner (RPC). This paper analyzes the composition and operation principles of this structure. To assess the economic benefits brought by the integration of photovoltaic and energy storage systems, a bilevel optimization model is established, with the objectives of optimizing energy storage capacity configuration and photovoltaic energy integration. The KKT (Karush–Kuhn–Tucker) method is employed to transform the model into a single-layer mixed-integer linear programming model, which is then solved using the CPLEX solver in MATLAB. The research findings indicate that, with the configuration of the ESS, the optimal PV consumption rate achieved is 96.8749%. Compared to a 100% PV consumption rate, the ESS capacity configuration is reduced by 13.14%, and the overall operational cost of the TPSS is at its lowest. The study suggests that the proposed bilevel optimization algorithm can more effectively consider PV consumption, leading to enhanced economic performance of the TPSS operation

OPTIMASI KEUNTUNGAN MENGGUNAKAN METODE KARUSH- KUHN-TUCKER (STUDI KASUS: MI ACEH PATTIMURA DI JAMBI)

Masalah pengeluaran yang tidak stabil dan produksi yang tidak optimal mendorong pelaku usaha untuk merumuskan strategi yang tepat agar usaha dapat terus berjalan dengan lancar. Hal tersebut juga terkait dengan adanya keinginan untuk memaksimalkan keuntungan. Masalah yang demikian dialami pula oleh pelaku usaha Mi Aceh Pattimura, Jambi. Adapun tujuan dari penelitian ini adalah untuk menentukan dan mengidentifikasi jumlah produksi yang optimal per hari supaya pelaku usaha Mi Aceh Pattimura memperoleh keuntungan harian yang optimal berdasarkan modal dan bahan yang tersedia. Dan pada penelitian ini, metode yang digunakan untuk menemukan keadaan optimal tersebut adalah metode Kuhn Tucker. Berdasarkan perhitungan menggunakan metode Karush-Kuhn-Tucker, diperoleh jumlah produksi optimal per hari pada usaha Mi Aceh Pattimura adalah mi aceh kuah sebanyak 15 porsi, mi aceh goreng sebanyak 6 porsi, mi aceh tumis sebanyak 19 porsi, mi aceh daging sebanyak 20 porsi, mi aceh ayam sebanyak 20 porsi, dan mi aceh udang sebanyak 20 porsi dengan keuntungan optimal yang dapat diperoleh sebesar Rp. 745.169,9279 per hari

Co-evolutionary Hybrid Bi-level Optimization

Multi-level optimization stems from the need to tackle complex problems involving multiple decision makers. Two-level optimization, referred as ``Bi-level optimization'', occurs when two decision makers only control part of the decision variables but impact each other (e.g., objective value, feasibility). Bi-level problems are sequential by nature and can be represented as nested optimization problems in which one problem (the ``upper-level'') is constrained by another one (the ``lower-level''). The nested structure is a real obstacle that can be highly time consuming when the lower-level is $\mathcal{NP}-hard$. Consequently, classical nested optimization should be avoided. Some surrogate-based approaches have been proposed to approximate the lower-level objective value function (or variables) to reduce the number of times the lower-level is globally optimized. Unfortunately, such a methodology is not applicable for large-scale and combinatorial bi-level problems. After a deep study of theoretical properties and a survey of the existing applications being bi-level by nature, problems which can benefit from a bi-level reformulation are investigated. A first contribution of this work has been to propose a novel bi-level clustering approach. Extending the well-know ``uncapacitated k-median problem'', it has been shown that clustering can be easily modeled as a two-level optimization problem using decomposition techniques. The resulting two-level problem is then turned into a bi-level problem offering the possibility to combine distance metrics in a hierarchical manner. The novel bi-level clustering problem has a very interesting property that enable us to tackle it with classical nested approaches. Indeed, its lower-level problem can be solved in polynomial time. In cooperation with the Luxembourg Centre for Systems Biomedicine (LCSB), this new clustering model has been applied on real datasets such as disease maps (e.g. Parkinson, Alzheimer). Using a novel hybrid and parallel genetic algorithm as optimization approach, the results obtained after a campaign of experiments have the ability to produce new knowledge compared to classical clustering techniques combining distance metrics in a classical manner. The previous bi-level clustering model has the advantage that the lower-level can be solved in polynomial time although the global problem is by definition $\mathcal{NP}$-hard. Therefore, next investigations have been undertaken to tackle more general bi-level problems in which the lower-level problem does not present any specific advantageous properties. Since the lower-level problem can be very expensive to solve, the focus has been turned to surrogate-based approaches and hyper-parameter optimization techniques with the aim of approximating the lower-level problem and reduce the number of global lower-level optimizations. Adapting the well-know bayesian optimization algorithm to solve general bi-level problems, the expensive lower-level optimizations have been dramatically reduced while obtaining very accurate solutions. The resulting solutions and the number of spared lower-level optimizations have been compared to the bi-level evolutionary algorithm based on quadratic approximations (BLEAQ) results after a campaign of experiments on official bi-level benchmarks. Although both approaches are very accurate, the bi-level bayesian version required less lower-level objective function calls. Surrogate-based approaches are restricted to small-scale and continuous bi-level problems although many real applications are combinatorial by nature. As for continuous problems, a study has been performed to apply some machine learning strategies. Instead of approximating the lower-level solution value, new approximation algorithms for the discrete/combinatorial case have been designed. Using the principle employed in GP hyper-heuristics, heuristics are trained in order to tackle efficiently the $\mathcal{NP}-hard$ lower-level of bi-level problems. This automatic generation of heuristics permits to break the nested structure into two separated phases: \emph{training lower-level heuristics} and \emph{solving the upper-level problem with the new heuristics}. At this occasion, a second modeling contribution has been introduced through a novel large-scale and mixed-integer bi-level problem dealing with pricing in the cloud, i.e., the Bi-level Cloud Pricing Optimization Problem (BCPOP). After a series of experiments that consisted in training heuristics on various lower-level instances of the BCPOP and using them to tackle the bi-level problem itself, the obtained results are compared to the ``cooperative coevolutionary algorithm for bi-level optimization'' (COBRA). Although training heuristics enables to \emph{break the nested structure}, a two phase optimization is still required. Therefore, the emphasis has been put on training heuristics while optimizing the upper-level problem using competitive co-evolution. Instead of adopting the classical decomposition scheme as done by COBRA which suffers from the strong epistatic links between lower-level and upper-level variables, co-evolving the solution and the mean to get to it can cope with these epistatic link issues. The ``CARBON'' algorithm developed in this thesis is a competitive and hybrid co-evolutionary algorithm designed for this purpose. In order to validate the potential of CARBON, numerical experiments have been designed and results have been compared to state-of-the-art algorithms. These results demonstrate that ``CARBON'' makes possible to address nested optimization efficiently