A dynamic gradient approach to Pareto optimization with nonsmooth convex objective functions

In a general Hilbert framework, we consider continuous gradient-like dynamical systems for constrained multiobjective optimization involving non-smooth convex objective functions. Our approach is in the line of a previous work where was considered the case of convex di erentiable objective functions. Based on the Yosida regularization of the subdi erential operators involved in the system, we obtain the existence of strong global trajectories. We prove a descent property for each objective function, and the convergence of trajectories to weak Pareto minima. This approach provides a dynamical endogenous weighting of the objective functions. Applications are given to cooperative games, inverse problems, and numerical multiobjective optimization

A matheuristic approach to solve the multi-objective beam angle optimisation problem in intensity modulated radiation therapy

Selecting a suitable set of beam angles is an important but difficult task in intensity modulated radiation therapy (IMRT) for cancer treatment. From a single objective point of view this problem, known as beam angle optimisation (BAO) problem, is solved by finding a beam angle configuration (BAC) that leads to the best dose distribution, according to some objective function. Because there exists a trade-off between the main goals in IMRT (to irradiate the tumour according to some prescription and to avoid surrounding healthy tissue) it makes sense to solve this problem from a multi-objective (MO) point of view. When doing so, a solution of the BAO problem is no longer a single BAC but instead a set of BACs which lead to a set of dose distributions that, depending on both dose prescription and physician preferences, can be selected as the preferred treatment. We solve this MO problem using a two-phase strategy. During the first phase, a deterministic local search algorithm is used to select a set of locally optimal BACs, according to a single objective function. During this search, an optimal dose distribution for each BAC, with respect to the single objective function, is calculated using an exact non-linear programming algorithm. During the second phase a set of non-dominated points is generated for each promising locally optimal BAC and a dominance analysis among them is performed. The output of the procedure is a set of (approximately) efficient BACs that lead to good dose distributions. To demonstrate the viability of the method, the two-phase strategy is applied to a prostate case

On multiobjective optimization from the nonsmooth perspective

Practical applications usually have multiobjective nature rather than having only one objective to optimize. A multiobjective problem cannot be solved with a single-objective solver as such. On the other hand, optimization of only one objective may lead to an arbitrary bad solutions with respect to other objectives. Therefore, special techniques for multiobjective optimization are vital. In addition to multiobjective nature, many real-life problems have nonsmooth (i.e. not continuously differentiable) structure. Unfortunately, many smooth (i.e. continuously differentiable) methods adopt gradient-based information which cannot be used for nonsmooth problems. Since both of these characteristics are relevant for applications, we focus here on nonsmooth multiobjective optimization. As a research topic, nonsmooth multiobjective optimization has gained only limited attraction while the fields of nonsmooth single-objective and smooth multiobjective optimization distinctively have attained greater interest. This dissertation covers parts of nonsmooth multiobjective optimization in terms of theory, methodology and application. Bundle methods are widely considered as effective and reliable solvers for single-objective nonsmooth optimization. Therefore, we investigate the use of the bundle idea in the multiobjective framework with three different methods. The first one generalizes the single-objective proximal bundle method for the nonconvex multiobjective constrained problem. The second method adopts the ideas from the classical steepest descent method into the convex unconstrained multiobjective case. The third method is designed for multiobjective problems with constraints where both the objectives and constraints can be represented as a difference of convex (DC) functions. Beside the bundle idea, all three methods are descent, meaning that they produce better values for each objective at each iteration. Furthermore, all of them utilize the improvement function either directly or indirectly. A notable fact is that none of these methods use scalarization in the traditional sense. With the scalarization we refer to the techniques transforming a multiobjective problem into the single-objective one. As the scalarization plays an important role in multiobjective optimization, we present one special family of achievement scalarizing functions as a representative of this category. In general, the achievement scalarizing functions suit well in the interactive framework. Thus, we propose the interactive method using our special family of achievement scalarizing functions. In addition, this method utilizes the above mentioned descent methods as tools to illustrate the range of optimal solutions. Finally, this interactive method is used to solve the practical case studies of the scheduling the final disposal of the spent nuclear fuel in Finland.Käytännön optimointisovellukset ovat usein luonteeltaan ennemmin moni- kuin yksitavoitteisia. Erityisesti monitavoitteisille tehtäville suunnitellut menetelmät ovat tarpeen, sillä monitavoitteista optimointitehtävää ei sellaisenaan pysty ratkaisemaan yksitavoitteisilla menetelmillä eikä vain yhden tavoitteen optimointi välttämättä tuota mielekästä ratkaisua muiden tavoitteiden suhteen. Monitavoitteisuuden lisäksi useat käytännön tehtävät ovat myös epäsileitä siten, etteivät niissä esiintyvät kohde- ja rajoitefunktiot välttämättä ole kaikkialla jatkuvasti differentioituvia. Kuitenkin monet optimointimenetelmät hyödyntävät gradienttiin pohjautuvaa tietoa, jota ei epäsileille funktioille ole saatavissa. Näiden molempien ominaisuuksien ollessa keskeisiä sovelluksia ajatellen, keskitytään tässä työssä epäsileään monitavoiteoptimointiin. Tutkimusalana epäsileä monitavoiteoptimointi on saanut vain vähän huomiota osakseen, vaikka sekä sileä monitavoiteoptimointi että yksitavoitteinen epäsileä optimointi erikseen ovat aktiivisia tutkimusaloja. Tässä työssä epäsileää monitavoiteoptimointia on käsitelty niin teorian, menetelmien kuin käytännön sovelluksien kannalta. Kimppumenetelmiä pidetään yleisesti tehokkaina ja luotettavina menetelminä epäsileän optimointitehtävän ratkaisemiseen ja siksi tätä ajatusta hyödynnetään myös tässä väitöskirjassa kolmessa eri menetelmässä. Ensimmäinen näistä yleistää yksitavoitteisen proksimaalisen kimppumenetelmän epäkonveksille monitavoitteiselle rajoitteiselle tehtävälle sopivaksi. Toinen menetelmä hyödyntää klassisen nopeimman laskeutumisen menetelmän ideaa konveksille rajoitteettomalle tehtävälle. Kolmas menetelmä on suunniteltu erityisesti monitavoitteisille rajoitteisille tehtäville, joiden kohde- ja rajoitefunktiot voidaan ilmaista kahden konveksin funktion erotuksena. Kimppuajatuksen lisäksi kaikki kolme menetelmää ovat laskevia eli ne tuottavat joka kierroksella paremman arvon jokaiselle tavoitteelle. Yhteistä on myös se, että nämä kaikki hyödyntävät parannusfunktiota joko suoraan sellaisenaan tai epäsuorasti. Huomattavaa on, ettei yksikään näistä menetelmistä hyödynnä skalarisointia perinteisessä merkityksessään. Skalarisoinnilla viitataan menetelmiin, joissa usean tavoitteen tehtävä on muutettu sopivaksi yksitavoitteiseksi tehtäväksi. Monitavoiteoptimointimenetelmien joukossa skalarisoinnilla on vankka jalansija. Esimerkkinä skalarisoinnista tässä työssä esitellään yksi saavuttavien skalarisointifunktioiden perhe. Yleisesti saavuttavat skalarisointifunktiot soveltuvat hyvin interaktiivisten menetelmien rakennuspalikoiksi. Täten kuvaillaan myös esiteltyä skalarisointifunktioiden perhettä hyödyntävä interaktiivinen menetelmä, joka lisäksi hyödyntää laskevia menetelmiä optimaalisten ratkaisujen havainnollistamisen apuna. Lopuksi tätä interaktiivista menetelmää käytetään aikatauluttamaan käytetyn ydinpolttoaineen loppusijoitusta Suomessa

Multi-objective optimisation: algorithms and application to computer-aided molecular and process design

Computer-Aided Molecular Design (CAMD) has been put forward as a powerful and systematic technique that can accelerate the identification of new candidate molecules. Given the benefits of CAMD, the concept has been extended to integrated molecular and process design, usually referred to as Computer-Aided Molecular and Process Design (CAMPD). In CAMPD approaches, not only is the interdependence between the properties of the molecules and the process performance captured, but it is also possible to assess the optimal overall performance of a given fluid using an objective function that may be based on process economics, energy efficiency, or environmental criteria. Despite the significant advances made in the field of CAM(P)D, there are remaining challenges in handling the complexities arising from the large mixed-integer nonlinear structure-property and process models and the presence of conflicting performance criteria that cannot be easily merged into a single metric. Many of the algorithms proposed to date, however, resort to single-objective decomposition-based approaches. To overcome these challenges, a novel CAMPD optimisation framework is proposed, in the first part of thesis, in the context of identifying optimal amine solvents for carbon dioxide (CO2) chemical absorption. This requires development and validation of a model that enables the prediction of process performance metrics for a wide range of solvents for which no experimental data exist. An equilibrium-stage model that incorporates the SAFT-γ Mie group contribution approach is proposed to provide an appropriate balance between accuracy and predictive capability with varying molecular design spaces. In order to facilitate the convergence behaviour of the process-molecular model, a tailored initialisation strategy is established based on the inside-out algorithm. Novel feasibility tests that are capable of recognising infeasible regions of molecular and process domains are developed and incorporated into an outer-approximation framework to increase solution robustness. The efficiency of the proposed algorithm is demonstrated by applying it to the design of CO2 chemical absorption processes. The algorithm is found to converge successfully in all 150 runs carried out. To derive greater insights into the interplay between solvent and process performance, it is desirable to consider multiple objectives. In the second part of the thesis, we thus explore the relative performance of five multi-objective optimisations (MOO) solution techniques, modified from the literature to address nonconvex MINLPs, on CAM(P)D problems to gain a better understanding of the performance of different algorithms in identifying the Pareto front efficiently. The combination of the sandwich algorithm with a multi-level single-linkage algorithm to solve nonconvex subproblems is found to perform best on average. Next, a robust algorithm for bi-objective optimisation (BOO), the SDNBI algorithm, is designed to address the theoretical and numerical challenges associated with the solution of general nonconvex and discrete BOO problems. The main improvements in the development of the algorithm are focused on the effective exploration of the nonconvex regions of the Pareto front and the early identification of regions where no additional Pareto solutions exist. The performance of the algorithm is compared to that of the sandwich algorithm and the modified normal boundary intersection method (mNBI) over a set of literature benchmark problems and molecular design problems. The SDNBI found to provide the most evenly distributed approximation of the Pareto front as well as useful information on regions of the objective space that do not contain a nondominated point. The advances in this thesis can accelerate the discovery of novel solvents for CO2 capture that can achieve improved process performance. More broadly, the modelling and algorithmic development presented extend the applicability of CAMPD and MOO based CAMD/CAMPD to a wider range of applications.Open Acces

Multi-objective optimisation methods applied to complex engineering systems

This research proposes, implements and analyses a novel framework for multiobjective optimisation through evolutionary computing aimed at, but not restricted to, real-world problems in the engineering design domain. Evolutionary algorithms have been used to tackle a variety of non-linear multiobjective optimisation problems successfully, but their success is governed by key parameters which have been shown to be sensitive to the nature of the particular problem, incorporating concerns such as the number of objectives and variables, and the size and topology of the search space, making it hard to determine the best settings in advance. This work describes a real-encoded multi-objective optimising evolutionary algorithm framework, incorporating a genetic algorithm, that uses self-adaptive mutation and crossover in an attempt to avoid such problems, and which has been benchmarked against both standard optimisation test problems in the literature and a real-world airfoil optimisation case. For this last case, the minimisation of drag and maximisation of lift coefficients of a well documented standard airfoil, the framework is integrated with a freeform deformation tool to manage the changes to the section geometry, and XFoil, a tool which evaluates the airfoil in terms of its aerodynamic efficiency. The performance of the framework on this problem is compared with those of two other heuristic MOO algorithms known to perform well, the Multi-Objective Tabu Search (MOTS) and NSGA-II, showing that this framework achieves better or at least no worse convergence. The framework of this research is then considered as a candidate for smart (electricity) grid optimisation. Power networks can be improved in both technical and economical terms by the inclusion of distributed generation which may include renewable energy sources. The essential problem in national power networks is that of power flow and in particular, optimal power flow calculations of alternating (or possibly, direct) current. The aims of this work are to propose and investigate a method to assist in the determination of the composition of optimal or high-performing power networks in terms of the type, number and location of the distributed generators, and to analyse the multi-dimensional results of the evolutionary computation component in order to reveal relationships between the network design vector elements and to identify possible further methods of improving models in future work. The results indicate that the method used is a feasible one for the achievement of these goals, and also for determining optimal flow capacities of transmission lines connecting the bus bars in the network