Pekka Malo

Finding optimal strategies in a multi-period multi-leader-follower Stackelberg game using an evolutionary algorithm

Stackelberg games are a classic example of bilevel optimization problems, which are often encountered in game theory and economics. These are complex problems with a hierarchical structure, where one optimization task is nested within the other. Despite a number of studies on handling bilevel optimization problems, these problems still remain a challenging territory, and existing methodologies are able to handle only simple problems with few variables under assumptions of continuity and differentiability. In this paper, we consider a special case of a multi-period multi-leader-follower Stackelberg competition model with non-linear cost and demand functions and discrete production variables. The model has potential applications, for instance in aircraft manufacturing industry, which is an oligopoly where a few giant firms enjoy a tremendous commitment power over the other smaller players. We solve cases with different number of leaders and followers, and show how the entrance or exit of a player affects the profits of the other players. In the presence of various model complexities, we use a computationally intensive nested evolutionary strategy to find an optimal solution for the model. The strategy is evaluated on a test-suite of bilevel problems, and it has been shown that the method is successful in handling difficult bilevel problems.

Multi-objective Stackelberg game between a regulating authority and a mining company: A case study in environmental economics

Bilevel programming problems are often found in practice. In this paper, we handle one such bilevel application problem from the domain of environmental economics. The problem is a Stakelberg game with multiple objectives at the upper level, and a single objective at the lower level. The leader in this case is the regulating authority, and it tries to maximize its total tax revenue over multiple periods while trying to minimize the environmental damages caused by a mining company. The follower is the mining company whose sole objective is to maximize its total profit over multiple periods under the limitations set by the leader. The solution to the model contains the optimal taxation and extraction decisions to be made by the players in each of the time periods. We construct a simplistic model for the Stackelberg game and provide an analytical solution to the problem. Thereafter, the model is extended to incorporate realism and is solved using a bilevel evolutionary algorithm capable of handling multiple objectives.

An improved bilevel evolutionary algorithm based on Quadratic Approximations

In this paper, we provide an improved evolutionary algorithm for bilevel optimization. It is an extension of a recently proposed Bilevel Evolutionary Algorithm based on Quadratic Approximations (BLEAQ). Bilevel optimization problems are known to be difficult and computationally demanding. The recently proposed BLEAQ approach has been able to bring down the computational expense significantly as compared to the contemporary approaches. The strategy proposed in this paper further improves the algorithm by incorporating archiving and local search. Archiving is used to store the feasible members produced during the course of the algorithm that provide a larger pool of members for better quadratic approximations of optimal lower level solutions. Frequent local searches at upper level supported by the quadratic approximations help in faster convergence of the algorithm. The improved results have been demonstrated on two different sets of test problems, and comparison results against the contemporary approaches are also provided.

Test Problem Construction for Single-Objective Bilevel Optimization

In this paper, we propose a procedure for designing controlled test problems for single-objective bilevel optimization. The construction procedure is flexible and allows its user to control the different complexities that are to be included in the test problems independently of each other. In addition to properties that control the difficulty in convergence, the procedure also allows the user to introduce difficulties caused by interaction of the two levels. As a companion to the test problem construction framework, the paper presents a standard test suite of 12 problems, which includes eight unconstrained and four constrained problems. Most of the problems are scalable in terms of variables and constraints. To provide baseline results, we have solved the proposed test problems using a nested bilevel evolutionary algorithm. The results can be used for comparison, while evaluating the performance of any other bilevel optimization algorithm. The code related to the paper may be accessed from the website http://bilevel.org.

Evaluating multivariate GARCH models in the Nordic electricity markets

ABSTRACT This article considers a variety of specification tests for multivariate GARCH models that are used for dynamic hedging in electricity markets. The test statistics include the robust conditional moments tests for sign-size bias along with the recently introduced copula tests for an appropriate dependence structure. We consider this effort worthwhile, since quite often the tests of multivariate GARCH models are omitted and the models become selected ad hoc depending on the results they generate. Hedging performance comparisons, in terms of unconditional and conditional ex-post variance portfolio reduction, are conducted.

A bilevel optimization approach to automated parameter tuning

Many of the modern optimization algorithms contain a number of parameters that require tuning before the algorithm can be applied to a particular class of optimization problems. A proper choice of parameters may have a substantial effect on the accuracy and efficiency of the algorithm. Until recently, parameter tuning has mostly been performed using brute force strategies, such as grid search and random search. Guesses and insights about the algorithm are also used to find suitable parameters or suggest strategies to adjust them. More recent trends include the use of meta-optimization techniques. Most of these approaches are computationally expensive and do not scale when the number of parameters increases. In this paper, we propose that the parameter tuning problem is inherently a bilevel programming problem. Based on this insight, we introduce an evolutionary bilevel algorithm for parameter tuning. A few commonly used optimization algorithms (Differential Evolution and Nelder-Mead) have been chosen as test cases, whose parameters are tuned on a number of standard test problems. The bilevel approach is found to quickly converge towards the region of efficient parameters. The code for the proposed algorithm can be accessed from the website http://bilevel.org.

Reduced form modeling of limit order markets

This paper proposes a parametric approach for stochastic modeling of limit order markets. The models are obtained by augmenting classical perfectly liquid market models with a few additional risk factors that describe liquidity properties of the order book. The resulting models are easy to calibrate and to analyse using standard techniques for multivariate stochastic processes. Despite their simplicity, the models are able to capture several properties that have been found in microstructural analysis of limit order markets. Calibration of a continuous-time three-factor model to Copenhagen Stock Exchange data exhibits, for example, mean reversion in liquidity as well as the so-called crowding out effect, which influences subsequent mid-price moves. Our dynamic models are also well suited for analysing market resilience after liquidity shocks.

Evolutionary algorithm for bilevel optimization using approximations of the lower level optimal solution mapping

Bilevel optimization problems are a class of challenging optimization problems, which contain two levels of optimization tasks. In these problems, the optimal solutions to the lower level problem become possible feasible candidates to the upper level problem. Such a requirement makes the optimization problem difficult to solve, and has kept the researchers busy towards devising methodologies, which can efficiently handle the problem. Despite the efforts, there hardly exists any effective methodology, which is capable of handling a complex bilevel problem. In this paper, we introduce bilevel evolutionary algorithm based on quadratic approximations (BLEAQ) of optimal lower level variables with respect to the upper level variables. The approach is capable of handling bilevel problems with different kinds of complexities in relatively smaller number of function evaluations. Ideas from classical optimization have been hybridized with evolutionary methods to generate an efficient optimization algorithm for a wide class of bilevel problems. The performance of the algorithm has been evaluated on two sets of test problems. The first set is a recently proposed SMD test set, which contains problems with controllable complexities, and the second set contains standard test problems collected from the literature. The proposed method has been compared against three benchmarks, and the performance gain is observed to be significant. The codes related to the paper may be accessed from the website http://bilevel.org.

Good debt or bad debt: Detecting semantic orientations in economic texts

The use of robo‐readers to analyze news texts is an emerging technology trend in computational finance. Recent research has developed sophisticated financial polarity lexicons for investigating how financial sentiments relate to future company performance. However, based on experience from fields that commonly analyze sentiment, it is well known that the overall semantic orientation of a sentence may differ from that of individual words. This article investigates how semantic orientations can be better detected in financial and economic news by accommodating the overall phrase‐structure information and domain‐specific use of language. Our three main contributions are the following: (a) a human‐annotated finance phrase bank that can be used for training and evaluating alternative models; (b) a technique to enhance financial lexicons with attributes that help to identify expected direction of events that affect sentiment; and (c) a linearized phrase‐structure model for detecting contextual semantic orientations in economic texts. The relevance of the newly added lexicon features and the benefit of using the proposed learning algorithm are demonstrated in a comparative study against general sentiment models as well as the popular word frequency models used in recent financial studies. The proposed framework is parsimonious and avoids the explosion in feature space caused by the use of conventional n‐gram features.

Automated query learning with Wikipedia and genetic programming

Most of the existing information retrieval systems are based on bag-of-words model and are not equipped with common world knowledge. Work has been done towards improving the efficiency of such systems by using intelligent algorithms to generate search queries, however, not much research has been done in the direction of incorporating human-and-society level knowledge in the queries. This paper is one of the first attempts where such information is incorporated into the search queries using Wikipedia semantics. The paper presents Wikipedia-based Evolutionary Semantics (Wiki-ES) framework for generating concept based queries using a set of relevance statements provided by the user. The query learning is handled by a co-evolving genetic programming procedure. To evaluate the proposed framework, the system is compared to a bag-of-words based genetic programming framework as well as to a number of alternative document filtering techniques. The results obtained using Reuters newswire documents are encouraging. In particular, the injection of Wikipedia semantics into a GP-algorithm leads to improvement in average recall and precision, when compared to a similar system without human knowledge. A further comparison against other document filtering frameworks suggests that the proposed GP-method also performs well when compared with systems that do not rely on query-expression learning.