On the Karush-Kuhn-Tucker reformulation of the bilevel optimization problem

This paper is mainly concerned with the classical KKT reformulation and the primal KKT reformulation (also known as an optimization problem with generalized equation constraint (OPEC)) of the optimistic bilevel optimization problem. A generalization of the MFCQ to an optimization problem with operator constraint is applied to each of these reformulations, hence leading to new constraint qualifications (CQs) for the bilevel optimization problem. M- and S-type stationarity conditions tailored for the problem are derived as well. Considering the close link between the aforementioned reformulations, similarities and relationships between the corresponding CQs and optimality conditions are highlighted. In this paper, a concept of partial calmness known for the optimal value reformulation is also introduced for the primal KKT reformulation and used to recover the M-stationarity conditions.     

Finding the K best policies in a finite-horizon Markov decision process

Directed hypergraphs represent a general modelling and algorithmic tool, which have been successfully used in many different research areas such as artificial intelligence, database systems, fuzzy systems, propositional logic and transportation networks. However, modelling Markov decision processes using directed hypergraphs has not yet been considered.In this paper we consider finite-horizon Markov decision processes (MDPs) with finite state and action space and present an algorithm for finding the K best deterministic Markov policies. That is, we are interested in ranking the first K deterministic Markov policies in non-decreasing order using an additive criterion of optimality. The algorithm uses a directed hypergraph to model the finite-horizon MDP. It is shown that the problem of finding the optimal policy can be formulated as a minimum weight hyperpath problem and be solved in linear time, with respect to the input data representing the MDP, using different additive optimality criteria.     

From a Nonlinear, Nonconvex Variational Problem to a Linear, Convex Formulation

Abstract. We propose a general approach to deal with nonlinear, nonconvex variational problems based on a reformulation of the problem resulting in an optimization problem with linear cost functional and convex constraints. As a first step we explicitly explore these ideas to some one-dimensional variational problems and obtain specific conclusions of an analytical and numerical nature.     

Convex relaxations and MIQCQP reformulations for a class of cardinality-constrained portfolio selection problems

In this paper we investigate a class of cardinality-constrained portfolio selection problems. We construct convex relaxations for this class of optimization problems via a new Lagrangian decomposition scheme. We show that the dual problem can be reduced to a second-order cone program problem which is tighter than the continuous relaxation of the standard mixed integer quadratically constrained quadratic program (MIQCQP) reformulation. We then propose a new MIQCQP reformulation which is more efficient than the standard MIQCQP reformulation in terms of the tightness of the continuous relaxations. Computational results are reported to demonstrate the tightness of the SOCP relaxation and the effectiveness of the new MIQCQP reformulation.     

Exact computational approaches to a stochastic uncapacitated single allocation p-hub center problem

The stochastic uncapacitated single allocation p-hub center problem is an extension of the deterministic version which aims to minimize the longest origin-destination path in a hub and spoke network. Considering the stochastic nature of travel times on links is important when designing a network to guarantee the quality of service measured by a maximum delivery time for a proportion of all deliveries. We propose an efficient reformulation for a stochastic p-hub center problem and develop exact solution approaches based on variable reduction and a separation algorithm. We report numerical results to show effectiveness of our new reformulations and approaches by finding global solutions of small-medium sized problems. The combination of model reformulation and a separation algorithm is particularly noteworthy in terms of computational speed.     

From a Nonlinear, Nonconvex Variational Problem to a Linear, Convex Formulation

   Abstract. We propose a general approach to deal with nonlinear, nonconvex variational problems based on a reformulation of the problem resulting in an optimization problem with linear cost functional and convex constraints. As a first step we explicitly explore these ideas to some one-dimensional variational problems and obtain specific conclusions of an analytical and numerical nature.     

Data-Driven Robust Chance Constrained Problems: A Mixture Model Approach

This paper discusses the mixture distribution-based data-driven robust chance constrained problem. We construct a data-driven mixture distribution-based uncertainty set from the perspective of simultaneously estimating higher-order moments. Then, we derive a reformulation of the data-driven robust chance constrained problem. As the reformulation is not a convex programming problem, we propose new and tight convex approximations based on the piecewise linear approximation method. We establish the theoretical foundation for these approximations. Finally, numerical results show that the proposed approximations are practical and efficient.     

A framework and a mean-field algorithm for the local control of spatial processes

The Markov Decision Process (MDP) framework is a tool for the efficient modelling and solving of sequential decision-making problems under uncertainty. However, it reaches its limits when state and action spaces are large, as can happen for spatially explicit decision problems. Factored MDPs and dedicated solution algorithms have been introduced to deal with large factored state spaces. But the case of large action spaces remains an issue. In this article, we define graph-based Markov Decision Processes (GMDPs), a particular Factored MDP framework which exploits the factorization of the state space and the action space of a decision problem. Both spaces are assumed to have the same dimension. Transition probabilities and rewards are factored according to a single graph structure, where nodes represent pairs of state/decision variables of the problem. The complexity of this representation grows only linearly with the size of the graph, whereas the complexity of exact resolution grows exponentially. We propose an approximate solution algorithm exploiting the structure of a GMDP and whose complexity only grows quadratically with the size of the graph and exponentially with the maximum number of neighbours of any node. This algorithm, referred to as MF-API, belongs to the family of Approximate Policy Iteration (API) algorithms. It relies on a mean-field approximation of the value function of a policy and on a search limited to the suboptimal set of local policies. We compare it, in terms of performance, with two state-of-the-art algorithms for Factored MDPs: SPUDD and Approximate Linear Programming (ALP). Our experiments show that SPUDD is not generally applicable to solving GMDPs, due to the size of the action space we want to tackle. On the other hand, ALP can be adapted to solve GMDPs. We show that ALP is faster than MF-API and provides solutions of similar quality for most problems. However, for some problems MF-API provides significantly better policies, and in all cases provides a better approximation of the value function of approximate policies. These promising results show that the GMDP model offers a convenient framework for modelling and solving a large range of spatial and structured planning problems, that can arise in many different domains where processes are managed over networks: natural resources, agriculture, computer networks, etc.     

Sequential optimality conditions for cardinality-constrained optimization problems with applications

Computational Optimization and Applications - Recently, a new approach to tackle cardinality-constrained optimization problems based on a continuous reformulation of the problem was proposed....     

Linear,Integer, Separable and Fuzzy Programming Problems: A Unified Approach towards Reformulation

For mathematical programming (MP) to have greater impact as a decision tool, MP software systems must offer suitable support in terms of model communication and modelling techniques. In this paper, modelling techniques that allow logical restrictions to be modelled in integer programming terms are described, and their implications discussed. In addition, it is illustrated that many classes of non-linearities which are not variable separable may be, after suitable algebraic manipulation, put in a variable separable form. The methods of reformulating the fuzzy linear programming problem as a max-min problem is also introduced. It is shown that analysis of bounds plays a key role in the following four important contexts: model reduction, reformulation of logical restrictions as 0-1 mixed integer programmes, reformulation of non-linear programmes as variable separable programmes and reformulation of fuzzy linear programmes. It is observed that, as well as incorporating an interface between the modeller and the optimizer, there is a need to make available to the modeller software facilities which support the model reformulation techniques described here.     

A reformulation for the stochastic lot sizing problem with service-level constraints

We study the stochastic lot-sizing problem with service level constraints and propose an efficient mixed integer reformulation thereof. We use the formulation of the problem present in the literature as a benchmark, and prove that the reformulation has a stronger linear relaxation. Also, we numerically illustrate that it yields a superior computational performance. The results of our numerical study reveals that the reformulation can optimally solve problem instances with planning horizons over 200 periods in less than a minute.     

Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems

In this paper, we propose a Newton-type method for solving a semismooth reformulation of monotone complementarity problems. In this method, a direction-finding subproblem, which is a system of linear equations, is uniquely solvable at each iteration. Moreover, the obtained search direction always affords a direction of sufficient decrease for the merit function defined as the squared residual for the semismooth equation equivalent to the complementarity problem. We show that the algorithm is globally convergent under some mild assumptions. Next, by slightly modifying the direction-finding problem, we propose another Newton-type method, which may be considered a restricted version of the first algorithm. We show that this algorithm has a superlinear, or possibly quadratic, rate of convergence under suitable assumptions. Finally, some numerical results are presented. Supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. Supported in part by the Scientific Research Grant-in-Aid from the Ministry of Education, Science and Culture, Japan.     

First-order inertial algorithms involving dry friction damping

We present a Branch-and-Cut algorithm for a class of nonlinear chance-constrained mathematical optimization problems with a finite number of scenarios. Unsatisfied scenarios can enter a recovery mode. This class corresponds to problems that can be reformulated as deterministic convex mixed-integer nonlinear programming problems with indicator variables and continuous scenario variables, but the size of the reformulation is large and quickly becomes impractical as the number of scenarios grows. The Branch-and-Cut algorithm is based on an implicit Benders decomposition scheme, where we generate cutting planes as outer approximation cuts from the projection of the feasible region on suitable subspaces. The size of the master problem in our scheme is much smaller than the deterministic reformulation of the chance-constrained problem. We apply the Branch-and-Cut algorithm to the mid-term hydro scheduling problem, for which we propose a chance-constrained formulation. A computational study using data from ten hydroplants in Greece shows that the proposed methodology solves instances faster than applying a general-purpose solver for convex mixed-integer nonlinear programming problems to the deterministic reformulation, and scales much better with the number of scenarios.

     

Modelling impacts of cropping systems: Demands and solutions for DEX methodology

Decision modelling of diverse groups of problems makes different requirements to the modelling methodologies and software. We present an actual decision problem and the required characteristics of corresponding decision models. The problem is from agronomy and addresses the ecological and economic impacts of cropping systems, with the focus on the differences between cropping systems with conventional crops and the ones with genetically modified crops. We describe the extensions of an existing DEX qualitative multi-attribute modelling methodology, which were made to cope with the challenges of the problem. The extensions address general hierarchical structures, probabilistic utility functions and numerical values of basic attributes. A new, freely available software tool called proDEX was implemented to support the extended methodology. In this paper we describe the problem of cropping system assessment, propose methodological extensions to DEX, and present the implementation of proDEX.     

Combining very large scale and ILP based neighborhoods for a two-level location problem

In this paper we tackle a generalization of the Single Source Capacitated Facility Location Problem in which two sets of facilities, called intermediate level and upper level facilities, have to be located; the dimensioning of the intermediate set, the assignment of clients to intermediate level facilities, and of intermediate level facilities to upper level facilities, must be optimized, as well. Such problem arises, for instance, in telecommunication network design: in fact, in hierarchical networks the traffic arising at client nodes often have to be routed through different kinds of facility nodes, which provide different services. We propose a heuristic approach, based on very large scale neighborhood search to tackle the problem, in which both ad hoc algorithms and general purpose solvers are applied to explore the search space. We report on experimental results using datasets from the capacitated location literature. Such results show that the approach is promising and that Integer Linear Programming based neighborhoods are significantly effective.     

Stochastic Nonlinear Complementarity Problems: Stochastic Programming Reformulation and Penalty-Based Approximation Method

We consider a class of stochastic nonlinear complementarity problems. We first reformulate the stochastic complementarity problem as a stochastic programming model. Based on the reformulation, we then propose a penalty-based sample average approximation method and prove its convergence. Finally, we report on some numerical test results to show the efficiency of our method.     

Interactive outranking approaches for multicriteria decision-making problems with imprecise information

PROMETHEE is a powerful method, which can solve many multiple criteria decision making (MCDM) problems. It involves sophisticated preference modelling techniques but requires too much a priori precise information about parameter values (such as criterion weights and thresholds). In this paper, we consider a MCDM problem where alternatives are evaluated on several conflicting criteria, and the criterion weights and/or thresholds are imprecise or unknown to the decision maker (DM). We build robust outranking relations among the alternatives in order to help the DM to rank the alternatives and select the best alternative. We propose interactive approaches based on PROMETHEE method. We develop a decision aid tool called INTOUR, which implements the developed approaches.     

A Simply Constrained Optimization Reformulation of KKT Systems Arising from Variational Inequalities

The Karush—Kuhn—Tucker (KKT) conditions can be regarded as optimality conditions for both variational inequalities and constrained optimization problems. In order to overcome some drawbacks of recently proposed reformulations of KKT systems, we propose casting KKT systems as a minimization problem with nonnegativity constraints on some of the variables. We prove that, under fairly mild assumptions, every stationary point of this constrained minimization problem is a solution of the KKT conditions. Based on this reformulation, a new algorithm for the solution of the KKT conditions is suggested and shown to have some strong global and local convergence properties. Accepted 10 December 1997     

Partial convexification cuts for 0–1 mixed-integer programs

In this research, we propose a new cut generation scheme based on constructing a partial convex hull representation for a given 0–1 mixed-integer programming problem by using the reformulation–linearization technique (RLT). We derive a separation problem that projects the extended space of the RLT formulation into the original space, in order to generate a cut that deletes a current fractional solution. Naturally, the success of such a partial convexification based cutting plane scheme depends on the process used to tradeoff the strength of the cut derived and the effort expended. Accordingly, we investigate several variable selection rules for performing this convexification, along with restricted versions of the accompanying separation problems, so as to be able to derive strong cuts within a reasonable effort. We also develop a strengthening procedure that enhances the generated cut by considering the binariness of the remaining unselected 0–1 variables. Finally, we present some promising computational results that provide insights into implementing the proposed cutting plane methodology.     

Constructing cyclic PBIBD(2)s through an optimization approach: Thirty‐two new cyclic designs

We formulate the construction of cyclic partially balanced incomplete block designs with two associate classes (PBIBD(2)s) as a combinatorial optimization problem. We propose an algorithm based on tabu search to tackle the problem. Our algorithm constructed 32 new cyclic PBIBD(2)s. © 2004 Wiley Periodicals, Inc.