History: Q-MIP and others

(Jan Wolf, Jan. 2017)

QLP / QIPi: In [194] a QLP formulation was used to model problems from the domain of real-time scheduling [57], with the goal to reduce the inflexibility of static scheduling in hard real-time systems. One of the fundamental features of real-time systems is the uncertain nature of execution times of tasks and the presence of complex timing constraints among them. In traditional models variable execution times are modeled through fixed worst-case values and relationships are limited to those that can be represented by precedence graphs [206]. Using QLPs as modeling tool allows to schedule tasks with varying execution times, represented by universally quantified variables, and complex timing constraints among tasks described by linear constraints.

In this context, a first polynomial time algorithm to solve QLPs based on quantifier elimination techniques was proposed in [108] for a restricted class of constraints [194], which are a subset of totally unimodular constraint systems [197]. For arbitrary linear constraints the worst-case time complexity of the proposed algorithm is double exponential [197, 125]. Subramani proposed a framework to formalize scheduling problems in real-time systems using QLPs in [207]. There he focussed on the variability in the execution times, complex relationships between tasks using arbitrary linear constraints and different scheduling schemes – static (98-QLPs), co-static (89-QLPs) or parametric (alternating quantifier changes) – depending on the information available when dispatching. A detailed introduction to QLPs and methodologies to solve them are available in [209, 147, 88]. QIPs were studied in [210], where a number of special cases where analyzed from the perspective of computational complexity. In that paper the authors discussed conditions under which QIPs can be relaxed to QLPs without altering the solution space.

Research in this context was also supported by the Deutsche Forschungsgemeinschaft (DFG). The DFG founded project “Erweiterung mathemtische Optimierungsmethoden zur Lösung PSPACE-vollständiger Probleme mit Hilfe quantifizierter linearer Programme” at TU Darmstadt dealt with the development of algorithms to solve QIPs based on extended LP- and QLP-techniques. The DFG founded Collaborative Research Centre (SFB) 805 “Sonderforschungsbereich 805 - Beherrschung von Unsicherheit in lasttragenden Systemen des Maschinenbaus” at TU Darmstadt deals with the control of uncertainty in mechanical systems. In this context Q-MIPs have been used to model and optimize process chains under uncertainty. Also in the context of game modeling QIPs can be applied. In [89] the PSPACE-complete two-person game Gomoku was modeled with the help of an QIP formulation, in [147] the model was adopted to the PSPACE-complete two-person game Connect-6 resulting in a 0/1-QIP that was solved with an extended variant of the algorithm proposed in [88].

Recently, extensions of traditional QLPs were presented in [91], where the standard quantification was extended to implications of quantified linear systems, called quantified linear implications (QLI). QLPs and QLIs that arise when universally quantified variables are unbounded were studied in [185]. Whereas in the above citations the focus was on analyzing various special cases of the general problem, with a view on subclasses that are tractable, our focus lies on general QLPs with two or multiple stages without any restrictions to the quantification sequence or the constraint system. Furthermore, our focus lies on QLP optimization problems, whereas the above citations consider quantified decision problems in general.

Optimization Under Uncertainty has experienced rapid development in both theory and practice from the very beginning in the 1950s, starting with the seminal works of Beale [15], Dantzig [72], Bellman [16] as well as Charnes and Cooper [65]. The particular importance of optimization under uncertainty was also demonstrated in the seminal paper by Ben-Tal and Nemirovski [23], where they showed in a detailed computational study that even a small perturbation of the problem data can make the nominal optimal solution completely meaningless from a practical viewpoint. We quote from their case study:

“In real-world applications of linear programming, one cannot ignore the possibility that a small uncertainty in the data can make the usual optimal solution completely meaningless from a practical viewpoint.”

Due to limited computational capabilities in the past, decision models often replaced those uncertainties with averages or best estimates but recent computational advantages greatly expanded the range of optimization techniques under uncertainty. Today stochastic programming and robust optimization – two fundamentally different approaches that address optimization with data uncertainty – are the most prominent modeling paradigms and presently the subject of intense research in this field. The main difference lies in the type of uncertainty that is handled by the two methodologies. Stochastic programming assumes the uncertainty to follow a probability distribution, while robust optimization in contrast is concerned with models that contain deterministic set-based uncertainties [36]. In the following we give a brief introduction to stochastic programming and robust optimization and compare them with quantified linear programming.

Stochastic Programming The approach for uncertainty quantification used in stochastic programming (see, e.g. [128, 174, 47, 189] and the references therein) is the representation of random parameters in the input data (c, A, b) by random variables. This approach has a long and active history dating at least as far back as Dantzigs original paper [72] and draws upon a variety of traditional operations research techniques. During the last four decades a vast quantity of literature on the subject has appeared (see, e.g. [218, 217] for a stochastic programming bibliography maintained by Maarten van der Vlerk). A variety of stochastic programming applications can be found in [224]. The underlying optimization problem can be a linear program, a (mixed) integer program, but also a nonlinear program. Usually, stochastic programming models are furthermore subdivided in recourse models and chance constraint models.

Recourse Models The more important and widely studied case, where most of the applications reported in the literature belong to and which has strong similarities to QLPs, is that of a stochastic linear program with recourse, which first appeared in Beale [15] and Dantzig [72]. These problems arise when some of the decision variables must be taken before the outcomes of some (or all) random events are known, and some after they become known. Recourse variables represent decisions that can be made on a “wait and see” basis, after the uncertainty is resolved, and they allow to adapt a solution to the specific outcome observed. Stochastic programming problems with recourse can be essentially divided into two-stage and multi-stage problems (TSSLPs and MSSLPs). In the former, a set of initial decisions are taken first, followed by a random event. After this, recourse decisions, which are based on this event, are taken. The multi-stage problem, accordingly consists of multiple stages, with a random event occurring between each stage.

Chance Constraints Models The essence of recourse models is that infeasibilities in the second or higher stages are allowed and can be compensated at a certain penalty. In chance constraint models, developed by Charnes and Cooper [65], the focus lies on the reliability of the system in contrast. This reliability is expressed as a minimum requirement on the probability of satisfying constraints. The resulting program is called a chance-constrained stochastic program. Under certain assumptions on the probability distribution (like e.g. Normal, Cauchy, Uniform), chance- constraints can be converted into a deterministic equivalent form and then be treated as in a traditional recourse problem as mentioned above. More details on chance-constrained stochastic programming can be found in [174].

Over the last 20 years, also stochastic programming with integer or binary recourse variables has received tremendous research attention in both application and algorithmic aspects. However, compared to the continuous case of a TSSLP or MSSLP, relatively little is known on theory and algorithms for linear mixed-integer stochastic programming in general. Given that duality results do not hold in general IP and MIP formulations, it is also difficult to extend current algorithms for the linear case in order to generate cutting planes, except when only the first stage contains integer variables or other special cases. A survey of developments can be found in [198, 199], a stochastic integer programming bibliography with publications from the years 1996-2007 can be found at [217].

Compared to stochastic programming models, a QLP can be interpreted as a worst-case MSSLP with fixed recourse and only the right-hand side being affected by uncertainties. The main advantage of quantified linear and integer programming is that uncertainty can be easily modeled without detailed knowledge of the underling probability distribution. Furthermore, while SP models minimize the expected value of an objective function with respect to a set of scenarios, QLPs minimize an objective function with respect to the possible worst case (maximum loss). There is a recursive node-based definition for QLPs, similar as for MSSLPs, and it has been shown how a DEP of a QLP can be constructed.

Robust Optimization In contrast to stochastic programming, robust optimization is a more recent approach in the context of optimization under uncertainty, in which the uncertainty model is not stochastic, but rather deterministic and set-based. Instead of seeking to immunize the solution in some probabilistic sense to stochastic uncertainty, here the decision-maker constructs a solution that is valid for any realization of uncertainty in a given set [36]. While applications of stochastic programming have been reported over many years in the literature, robust optimization appeared recently, with most research in the past ten years (see, e.g. [24, 38, 42, 18, 36, 215] and the references therein). The roots can be found in the field of robust control and in the work of Soyster [204]. However, high interest in both theoretical aspects and practical applications started in the first place with the work of Ben-Tal and Nemirovski [20, 21], and El-Ghaoui et al [109, 110] in the late 1990s.

The idea of robust optimization is to define a so-called uncertanty set U for the uncertain parameters in the input data (c, A, b) and then to require that the constraint system should hold for any possible realization of the data from U. The optimization problem modeling this requirement is called the robust counterpart problem (RCP), an optimal solution of the RCP is called robust optimal solution. The robust optimization methodology can be applied to any optimization problem (e.g. quadratically constrained quadratic programs (QCQPs) or semidefinite programs (SDPs)), in which the uncertain data is known to lie in a given uncertainty set. Though it is still a relatively new approach, it has already proved very useful in many applications. In [18] a list of problem classes for which robust optimization is applicable is listed.

One might imagine that the addition of robustness to a general optimization problem comes at the expense of a significantly increased computational complexity, and indeed, the RCP of an arbitrary convex optimization problem is in general intractable [18]. Nevertheless, depending on the structure of the underlying optimization problem and the uncertainty set U , the corresponding RCP can be solved or at least be approximated in polynomial time for many application cases [21]. Ben-Tal and Nemirovski [22] showed that the RCP of a linear optimization problem is essentially always tractable for many practical uncertainty sets.

Due to the results in [18], in the case of robust linear optimization, the objective and the right-hand side of the constraints can be assumed to be deterministic and w.l.o.g. constraint-wise uncertainty can be assumed. It hence suffices to focus on single constraints of a special form. These are referred as the subproblems which must be solved in order to solve the entire RCP. Of course, a major modeling decision concerns the choice of the uncertainty set U and the resulting RCP might not longer be an LP [17]. In particular, the RCP of an LP with a polyhedral uncertainty set becomes a linear programming problem, while the RCP of an LP with an ellipsoidal uncertainty set becomes a second-order cone problem respectively [24]. Other interesting results have been achieved recently, e.g. under very natural assumptions, robust LPs can be formulated as semi-definite programming problems and thus solved by a polynomial time approximation scheme [21, 18]. Several attempts to extend the ideas to discrete robust optimization have been e.g. made in [37, 40, 39]. However, unlike the case of continuous robust optimization, the conditions on the uncertainty sets are rather restrictive to still ensure the efficient solvability of the RCP as shown in [39]. Therefore, approximation techniques based on piecewise linearization are currently the means of choice for such problems [42].

While stochastic programming problems can result in large deterministic linear models when considering many scenarios, the RCP models grow only slightly when uncertainty is added in general and therefore, they can often be solved efficiently. Another crucial difference is that in the stochastic programming approach constraints may be violated with a certain probability (chance-constraints) or at a given penalty (recourse model), whereas robust optimization is associated with hard constraints, which must be satisfied whatever the realization of the data (c, A, b) in the uncertainty set U is [22]. Thus, in many cases a single-stage robust optimization model tends to be to conservative [11], especially in situations where some recourse decisions can be made after the uncertainty is revealed. To address this issue, two-stage models – also called robust adjustable or adaptable optimization – were proposed e.g. in [19]. However, two-stage problems are very challenging to compute, even formulations with LP problems in both stages can be NP-hard [19]. Nevertheless, in many real-world problems, decisions and uncertainty are often repeatedly interwoven over time and RO model are not capable to handle this efficiently due to the difficulty in incorporating multiple stages [215].

From the viewpoint of quantified linear and integer programming, robust optimization problems can be viewed as QLPs with one quantifier change (98), whereas in QLPs multiple quantifier changes occur in general. However, from the viewpoint of robust optimization, where arbitrary uncertainty sets may appear (though not necessarily being tractable at all), the uncertainty set of QLPs is a simple hyperrectangle.

Other LP-based solution techniques Apart from stochastic programming and robust optimization as mentioned above, typical solution approaches for problems that are affected by uncertainty are sensitivity analysis [179], LP-based approximation techniques [155], dynamic programming [16], and the exploration of Markov-chains [237] for example. Sensitivity analysis is a simple classical method of addressing uncertainty, which is typically applied as a post-optimization tool for quantifying the goodness or robustness of a solution, e.g. the change in cost for small perturbations in the underlying problem data. It is not particularly helpful for generating solutions that are robust to data changes in the sense of an a priori ensured feasibility when the problem parameters vary within the prescribed uncertainty set [234]. A further possibility to deal with these uncertainties is an approximation where a given probability distribution is aggregated to a single estimated number. Then, the optimum concerning these estimated input data can be computed with the help of traditional optimization tools. In some fields of application, as e.g. the fleet assignment problem of airlines, this procedure was successfully established [117]. In other fields, like production planning and control, this technique could not be successfully applied, although mathematical models do exist [172]. Dynamic programming was developed by Richard Bellman in the early1950s [16] and is a computational method to deal with multi-stage decision processes in a dynamic environment. In fact, decision-making under uncertainty was the original and intended application of this technique [164]. Dynamic programming usually refers to simplifying a decision by breaking it down into a sequence of decision steps over time. It starts solving all subproblems at the final stage and then uses the solution to recursively compute solutions for higher stages until the original problem at the root is solved eventually. For more information on solution approaches in the context of optimization under uncertainty we refer the reader to [193, 41] and the references therein.

Quantification of Variables and Constraints The explicit quantification of variables and constraints allows to express many problems that cannot be modeled using classical modeling paradigms. Quantifiers are a powerful tools to model uncertainty or adversarial situations. For example, if a decision is based on a value that is not exactly known in advance, one might look for robust solutions that are valid for all possible values that might occur. As the efforts of extending languages with quantifiers have not only been made for linear programs in terms of QLPs and QIPs, we give an overview in the following, where we briefly sketch the achievements from other fields in this context.

Boolean Satisfiablility (SAT) and Constraint Satisfaction (CSP) Many real-world problems are combinatorial search problems that can be represented in terms of decision variables and constraints. Besides integer programming (IP), boolean satisfiability problems (SAT) and constraint satisfaction problems (CSP) are very successful frameworks that are used to model and solve these problems.

The SAT problem is one of the most important and extensively studied problems in computer science. Given a propositional boolean formula, the SAT problem asks for an assignment of variables such that the formula evaluates to true, or a proof that no such assignment exists [10, 44]. SAT is one of the classic problems in computer science, it is of theoretical interest because it is the canonical NP-complete problem [70] (cf. Section 4.2 for details). Apart from its high theoretical relevance, SAT has many practical applications as e.g. model checking, design verification, or AI planning [152, 68].

Integer Programming (IP) generalizes the standard SAT problem, by allowing the possible values for the variables to be chosen from an arbitrary finite set, and allowing the constraints to be arbitrary linear constraints rather than just clauses. Whereas the solution approaches for IPs often rely on LP-relaxations combined with branch-and-cut procedures to find optimal integer solutions, the core of most modern SAT solvers is the Davis-Putnam-Logemann-Loveland (DPLL) algorithm [78, 77], which is essentially a depth-first-search with some pruning techniques like unit propagation [235], conflict-driven clause learning [153], and back- tracking [150].

The constraint satisfaction problem (CSP) also generalizes the standard SAT problem, by allowing the possible values for the variables to be chosen from an arbitrary finite set. Furthermore, the constraints are allowed to to be arbitrary constraints rather than just clauses or linear constraints and thus, CSP also generalizes IP. The CSP problem consists of a set of variables, each with a finite domain of values and a set of constraints on subsets of these variables.
CSPs provide a general framework to express a wide variety of combinatorial search problems and they play an important role in many areas of computer science and artificial intelligence [52]. They also find their application in the Operations Research community in the context of Location Planning, Scheduling, Car Sequencing, Cutting Stock Problems, Vehicle Routing, Timetabeling, Rostering and many more [54]. SAT, CSP and the IP feasibility problem are three different paradigms to model and solve combinatorial search problems, and as each of the problem classes is NP-complete, they can be reduced among each others in polynomial time [225, 32, 238]. However, each paradigm has its strengths and weaknesses. Due to their complementary strengths, there is an increasing belief that hybrid methods may often perform better than pure methods. An approach to integrate techniques from constraint programming and (mixed-integer) programming in order to exploit the strengths of both fields, is the constraint integer programming framework SCIP [2, 4]. However, there are classes of problems containing uncertainty that cannot be expressed within these frameworks. QCSPs and QSAT, which are the quantified extensions of CSP and SAT and allow for universally quantified variables, make it possible to model such problems that contain uncertainty. As a result, these extensions have been attracting significant interest in recent years. In the following, we highlight the key ideas of both approaches and specify their basic solution strategies.

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