B06
Outer approximation for generalized convex mixed-integer nonlinear robust optimization problems
(2024)
We consider nonlinear robust optimization problems with mixed-integer decisions as well as nonconvexities. In detail, we consider cases where objective and constraint functions can be nonsmooth and generalized convex, i.e., f°-quasiconvex or f°-pseudoconvex.
We propose an algorithm for such robust optimization problems that does not require a certain structure of the adversarial problem but only requires that approximate worst cases are available. As a result, our algorithm finds a robust optimal solution up to a tolerance. Our method integrates a bundle method into an outer approximation approach where the bundle method is used for the arising continuous subproblems. We rely on methods from the literature, namely a bundle method for nonlinear and nonconvex robust optimization problems and outer approximation approaches for quasiconvex settings. Our contribution is to combine them to one convergent robust optimization method that can cope with inexactness of worst-case evaluations.
Further, we propose the gas transport under uncertainties as a relevant application and demonstrate that generalized convexity is fulfilled for a type of a network structure.
A Gradient-Based Method for Joint Chance-Constrained Optimization with Continuous Distributions
(2024)
The input parameters of an optimization problem are often affected by uncertainties. Chance constraints are a common way to model stochastic uncertainties in the constraints. Typically, algorithms for solving chance-constrained problems require convex functions or discrete probability distributions. In this work, we go one step further and allow non-convexities as well as continuous distributions. We propose a gradient-based approach to approximately solve joint chance-constrained models. We approximate the original problem by smoothing indicator functions. Then, the smoothed chance constraints are relaxed by penalizing their violation in the objective function. The approximation problem is solved with the Continuous Stochastic Gradient method that is an enhanced version of the stochastic gradient descent and has recently been introduced in the literature. We present a convergence theory for the smoothing and penalty approximations. Under very mild assumptions, our approach is applicable to a wide range of chance-constrained optimization problems. As an example, we illustrate its computational efficiency on difficult practical problems arising in the operation of gas networks. The numerical experiments demonstrate that the approach quickly finds nearly feasible solutions for joint chance-constrained problems with non-convex constraint functions and continuous distributions, even for realistically-sized instances.
Stochastic Optimization (SO) is a classical approach for optimization under uncertainty that typically requires knowledge about the probability distribution of uncertain parameters. As the latter is often unknown, Distributionally Robust Optimization (DRO) provides a strong alternative that determines the best guaranteed solution over a set of distributions (ambiguity set). In this work, we present an approach for DRO over time that uses online learning and scenario observations arriving as a data stream to learn more about the uncertainty. Our robust solutions adapt over time and reduce the cost of protection with shrinking ambiguity. For various kinds of ambiguity sets, the robust solutions converge to the SO solution. Our algorithm achieves the optimization and learning goals without solving the DRO problem exactly at any step. We also provide a regret bound for the quality of the online strategy which converges at a rate of $ O(\log T / \sqrt{T})$, where $T$ is the number of iterations. Furthermore, we illustrate the effectiveness of our procedure by numerical experiments on mixed-integer optimization instances from popular benchmark libraries and give practical examples stemming from telecommunications and routing. Our algorithm is able to solve the DRO over time problem significantly faster than standard reformulations.
In many real-world mixed-integer optimisation problems from engineering, the side
constraints can be subdivided into two categories: constraints which describe a certain logic to model a feasible allocation of resources (such as a maximal number of available assets, working time requirements, maintenance requirements, contractual obligations, etc.),
and constraints which model physical processes and the related quantities (such as current,
pressure, temperature, etc.). While the first type of constraints can often easily be stated in
terms of a mixed-integer program (MIP), the second part may involve the incorporation of
complex non-linearities, partial differential equations or even a black-box simulation of the
involved physical process. In this work, we propose the integration of a trained tree-based
classifier – a decision-tree or a random forest, into a mixed-integer optimization model as a
possible remedy. We assume that the classifier has been trained on data points produced
by a detailed simulation of a given complex process to represent the functional relationship
between the involved physical quantities. We then derive MIP-representable reformulations
of the trained classifier such that the resulting model can be solved using state-of-the-art
solvers. At the hand of several use cases in terms of possible optimisation goals, we show
the broad applicability of our framework that is easily extendable to other tasks beyond
engineering. In a detailed real-world computational study for the design of stable direct-
current power networks, we demonstrate that our approach yields high-quality solutions
in reasonable computation times.
Robust DC Optimal Power Flow with Modeling of Solar Power Supply Uncertainty via R-Vine Copulas
(2021)
We present a robust approximation of joint chance constrained DC Optimal Power Flow in combination with a model-based prediction of uncertain power supply via R-vine copulas.
It is applied to optimize the discrete curtailment of solar feed-in in an electrical distribution network and guarantees network stability under fluctuating feed-in.
This is modeled by a two-stage mixed-integer stochastic optimization problem proposed by Aigner et al. (European Journal of Operational Research, (2021)).
The solution approach is based on the approximation of chance constraints via robust constraints using suitable uncertainty sets.
The resulting robust optimization problem has a known equivalent tractable reformulation.
To compute uncertainty sets that lead to an inner approximation of the stochastic problem, an R-vine copula model is fitted to the distribution of the multi-dimensional power forecast error, i.e., the difference between the forecasted solar power and the measured feed-in at several network nodes.
The uncertainty sets are determined by encompassing a sufficient number of samples drawn from the R-vine copula model.
Furthermore, an enhanced algorithm is proposed to fit R-vine copulas which can be used to draw conditional samples for given solar radiation forecasts.
The experimental results obtained for real-world weather and network data demonstrate the effectiveness of the combination of stochastic programming and model-based prediction of uncertainty via copulas.
We improve the outcomes of previous work by showing that the resulting uncertainty sets are much smaller and lead to less conservative solutions while maintaining the same probabilistic guarantees.
In a recent article the so called continuous stochastic gradient method (CSG) for the efficient solution of a class of stochastic optimization problems was introduced. While the applicability of known stochastic gradient type methods is typically limited to so called expected risk functions, no such limitation exists for CSG. The key to this lies in the computation of design dependent integration weights, which allows for an optimal usage of available information leading to stronger convergence properties. However, due to the nature of the formula for these integration weights, the practical applicability was essentially limited to problems, in which stochasticity enters via a low-dimensional and suficiently simple probability distribution. In this paper the scope of the CSG method is significantly extended presenting new ways of calculating the integration weights. A full convergence analysis for this new variant of the CSG method is presented and its efficiency is demonstrated in comparison to more classical stochastic gradient methods by means of a number of problem classes, relevant in stochastic optimization and machine learning.
This work studies robust gas network optimization under uncertainties in demand and in the physical parameters. The corresponding optimization problems are nonconvex in node pressures and flows along the pipes. They are thus very difficult to solve for realistic instance sizes. In recent approaches, an adaptive bundle method has been developed, where one solves the occurring adversarial problems via iteratively refined piecewise linear relaxations. These subproblems need to be solved always from scratch using mixed-integer linear programming (MIP). As alternative to the MIP solver, we employ here a nonsmooth optimization approach that allows a warm start strategy such that it can profit from the results obtained for coarser relaxations. We evaluate the approach for realistic gas network topologies and outline possibilities for future research.
Pareto efficiency for robust linear programs was introduced by
Iancu and Trichakis in [9]. We generalize their approach and theoretical results
to robust optimization problems in Euclidean spaces with linear uncertainty.
Additionally, we demonstrate the value of this approach in an exemplary
manner in the area of robust semidefinite programming (SDP). In particular,
we prove that computing a Pareto robustly optimal solution for a robust SDP
is tractable and illustrate the benefit of such solutions at the example of the
maximal eigenvalue problem. Furthermore, we modify the famous algorithm of
Goemans and Williamson [8] in order to compute cuts for the robust max cut
problem that yield an improved approximation guarantee in non-worst-case
scenarios.
We consider equilibrium problems under uncertainty where firms
maximize their profits in a robust way when selling their output. Robust
optimization plays an increasingly important role when best guaranteed objective
values are to be determined, independently of the specific distributional
assumptions regarding uncertainty. In particular, solutions are to be determined
that are feasible regardless of how the uncertainty manifests itself within
some predefined uncertainty set. Our analysis adopts the robust optimization
perspective in the context of equilibrium problems. First, we consider a singlestage,
nonadjustable robust setting. We then go one step further and study the
more complex two-stage or adjustable case where a part of the variables can
adjust to the realization of the uncertainty. We compare equilibrium outcomes
with the corresponding centralized robust optimization problem where the
sum of all profits are maximized. As we find, the market equilibrium for
the perfectly competitive firms differs from the solution of the robust central
planner, which is in stark contrast to classical results regarding the efficiency of
market equilibria with perfectly competitive firms. For the different scenarios
considered, we furthermore are able to determine the resulting price of anarchy.
In the case of non-adjustable robustness, for fixed demand in every time step
the price of anarchy is bounded whereas it is unbounded if the buyers are
modeled by elastic demand functions. For the two-stage adjustable setting,
we show how to compute subsidies for the firms that lead to robust welfare
optimal equilibria.
Currently, few approaches are available for mixed-integer nonlinear robust optimization. Those that do exist typically either require restrictive assumptions on the problem structure or do not guarantee robust protection. In this work, we develop an algorithm for convex mixed-integer nonlinear robust optimization problems where a key feature is that the method does not rely on a specific structure of the inner worst-case (adversarial) problem and allows the latter to be non-convex. A major challenge of such a general nonlinear setting is ensuring robust protection, as this calls for a global solution of the non-convex adversarial problem. Our method is able to achieve this up to a tolerance, by requiring worst-case evaluations only up to a certain precision. For example, the necessary assumptions can be met by approximating a non-convex adversarial via piecewise relaxations and solving the resulting problem up to any requested error as a mixed-integer linear problem.
In our approach, we model a robust optimization problem as a nonsmooth mixed-integer nonlinear problem and tackle it by an outer approximation method that requires only inexact function values and subgradients. To deal with the arising nonlinear subproblems, we render an adaptive bundle method applicable to this setting and extend it to generate cutting planes, which are valid up to a known precision. Relying on its convergence to approximate critical points, we prove, as a consequence, finite convergence of the outer approximation algorithm.
As an application, we study the gas transport problem under uncertainties in demand and physical parameters on realistic instances and provide computational results demonstrating the efficiency of our method.
We study the robust maximum flow problem and the robust maximum flow over time problem where a given number of arcs Γ may fail or may be delayed. Two prominent models have been introduced for these problems: either one assigns flow to arcs fulfilling weak flow conservation in any scenario, or one assigns flow to paths where an arc failure or delay affects a whole path. We provide a unifying framework by presenting novel general models, in which we assign flow to subpaths. These models contain the known models as special cases and unify their advantages in order to obtain less conservative robust solutions.
We give a thorough analysis with respect to complexity of the general models. In particular, we show that the general models are essentially NP-hard, whereas, e.g. in the static case with Γ=1 an optimal solution can be computed in polynomial time. Further, we answer the open question about the complexity of the dynamic path model for Γ=1. We also compare the solution quality of the different models. In detail, we show that the general models have better robust optimal values than the known models and we prove bounds on these gaps.
We present a solution framework for general alternating current optimal power flow (AC OPF) problems that include discrete decisions.
The latter occur, for instance, in the context of the curtailment of renewables or the
switching of power generation units and transmission lines.
Our approach delivers globally optimal solutions and is provably convergent.
We model AC OPF problems with discrete decisions as mixed-integer nonlinear programs.
The solution method starts from a known framework that uses piecewise linear relaxations.
These relaxations are modeled as as mixed-integer linear programs and adaptively refined until some termination criterion is fulfilled.
In this work, we extend and complement this approach by problem-specific as well as very general algorithmic enhancements.
In particular, these are mixed-integer second-order cone programs as well as primal and dual cutting planes.
For example objective cuts and no-good-cuts help to compute good feasible solutions as where outer approximation constraints tighten the relaxations.
We present extensive numerical results for various AC OPF problems where discrete decisions play a major role.
Even for hard instances with a large proportion of discrete decisions, the method is able
to generate high quality solutions efficiently.
Furthermore, we compare our approach with state-of-the-art MINLP.
Our method outperforms all other algorithms.
Linear complementarity problems are a powerful tool for modeling many practically relevant situations such as market equilibria. They also connect many sub-areas of mathematics like game theory, optimization, and matrix theory. Despite their close relation to optimization, the protection of LCPs against uncertainties - especially in the sense of robust optimization - is still in its infancy. During the last years, robust LCPs have only been studied using the notions of strict and Γ-robustness. Unfortunately, both concepts lead to the problem that the existence of robust solutions cannot be guaranteed. In this paper, we consider affinely adjustable robust LCPs. In the latter, a part of the LCP solution is allowed to adjust via a function that is affine in the uncertainty. We show that this notion of robustness allows to establish strong characterizations of solutions for the cases of uncertain matrix and vector, separately, from which existence results can be derived. Our main results are valid for the case of an uncertain LCP vector. Here, we additionally provide sufficient conditions on the LCP matrix for the uniqueness of a solution. Moreover, based on characterizations of the affinely adjustable robust solutions, we derive a mixed-integer programming formulation that allows to solve the corresponding robust counterpart. If, in addition, the certain LCP matrix is positive semidefinite, we prove polynomial-time solvability and uniqueness of robust solutions. If the LCP matrix is uncertain, characterizations of solutions are developed for every nominal matrix, i.e., these characterizations are, in particular, independent of the definiteness of the nominal matrix. Robust solutions are also shown to be unique for positive definite LCP matrix but both uniqueness and mixed-integer programming formulations still remain open problems if the nominal LCP matrix is not positive definite.
We propose a mathematical optimization model and its solution for joint chance constrained DC Optimal Power Flow. In this application, it is particularly important that there is a high probability of transmission limits being satisfied, even in the case of uncertain or fluctuating feed-in from renewable energy sources. In critical network situations where the network risks overload, renewable energy feed-in has to be curtailed by the transmission system operator (TSO). The TSO can reduce the feed-in in discrete steps at each network node. The proposed optimization model minimizes curtailment while ensuring that there is a high probability of transmission limits being maintained. The latter is modeled via (joint) chance constraints that are computationally challenging. Thus, we propose a solution approach based on the robust safe approximation of these constraints. Hereby, probabilistic constraints are replaced by robust constraints with suitably defined uncertainty sets constructed from historical data. The uncertainty sets are calculated by encompassing randomly drawn scenarios using the scenario approach proposed by Margellos et al. (IEEE Transactions on Automatic Control, 59 (2014)). The ability to discretely control the power feed-in then leads to a robust optimization problem with decision-dependent uncertainties, i.e. the uncertainty sets depend on decision variables. We propose an equivalent mixed-integer linear reformulation for box uncertainties with the exact linearization of bilinear terms. Finally, we present numerical results for different test cases from the Nesta archive, as well as for a real network. We consider the discrete curtailment of solar feed-in, for which we use real-world weather and network data. The experimental tests demonstrate the effectiveness of this method and run times are very fast. Moreover, on average the calculated robust solutions lead only to a small increase in curtailment, when compared to nominal solutions.
Currently, there are few theoretical or practical approaches available for general nonlinear robust optimization. Moreover, the approaches that do exist impose restrictive assumptions on the problem structure. We present an adaptive bundle method for nonlinear and non-convex robust optimization problems with a suitable notion of inexactness in function values and subgradients. As the worst case evaluation requires a global solution to the adversarial problem, it is a main challenge in a general non-convex nonlinear setting. Moreover, computing elements of an epsilon-perturbation of the Clarke subdifferential in the l2-norm sense is in general prohibitive for this class of problems. In this article, instead of developing an entirely new bundle concept, we demonstrate how existing approaches, such as Noll's bundle method for non-convex minimization with inexact information (Computational and analytical mathematics 50: 555-592, 2013) can be modified to be able to cope with this situation. Extending the non-convex bundle concept to the case of robust optimization in this way, we prove convergence under two assumptions: Firstly, that the objective function is lower C1 and secondly, that approximately optimal solutions to the adversarial maximization problem are available. The proposed method is hence applicable to a rather general setting of nonlinear robust optimization problems. In particular, we do not rely on a specific structure of the adversary's constraints. The considered class of robust optimization problems covers the case that the worst-case adversary only needs to be evaluated up to a certain precision. One possibility to evaluate the worst case with the desired degree of precision is the use of techniques from mixed-integer linear programming (MIP).
We investigate the procedure on some analytic examples. As applications, we study the gas transport problem under uncertainties in demand and in physical parameters that affect pressure losses in the pipes. Computational results for examples in large realistic gas network instances demonstrate the applicability as well as the efficiency of the method.
Solving mixed-integer nonlinear optimization problems (MINLPs) to global optimality is extremely challenging. An important step for enabling their solution consists in the design of convex relaxations of the feasible set. Known solution approaches based on spatial branch-and-bound become more effective the tighter the used relaxations are. Relaxations are commonly established by convex underestimators, where each constraint function is considered separately. Instead, a considerably tighter relaxation can be found via so-called simultaneous convexification, where convex underestimators are derived for more than one constraint function at a time. In this work, we present a global solution approach for solving mixed-integer nonlinear problems that uses simultaneous convexification. We introduce a separation method that relies on determining the convex envelope of linear combinations of the constraint functions and on solving a nonsmooth convex problem. In particular, we apply the method to quadratic absolute value functions and derive their convex envelopes. The practicality of the proposed solution approach is demonstrated on several test instances from gas network optimization, where the method outperforms standard approaches that use separate convex relaxations.
For a mixed-integer linear problem (MIP) with uncertain constraints, the radius of robust feasibility (RRF) determines a value for the maximal “size” of the uncertainty set such that robust feasibility of the MIP can be guaranteed. The approaches for the RRF in the literature are restricted to continuous optimization problems. We first analyze relations between the RRF of a MIP and its continuous linear (LP) relaxation. In particular, we derive conditions under which a MIP and its LP relaxation have the same RRF. Afterward, we extend the notion of the RRF such that it can be applied to a large variety of optimization problems and uncertainty sets. In contrast to the setting commonly used in the literature, we consider for every constraint a potentially different uncertainty set that is not necessarily full-dimensional. Thus, we generalize the RRF to MIPs as well as to include “safe” variables and constraints, i.e., where uncertainties do not affect certain variables or constraints. In the extended setting, we again analyze relations between the RRF for a MIP and its LP relaxation. Afterward, we present methods for computing the RRF of LPs as well as of MIPs with safe variables and constraints. Finally, we show that the new methodologies can be successfully applied to the instances in the MIPLIB 2017 for computing the RRF.
In the course of the energy transition, load and supply centers are growing apart in electricity markets worldwide, rendering regional price signals even more important to provide adequate locational investment incentives. This paper focuses on electricity markets that operate under a zonal pricing market design. For a fixed number of zones, we endogenously derive the optimal configuration of price zones and available transfer capacities on a network in order to optimally govern investment and production decisions in the long run. In a multilevel mixed-integer nonlinear model that contains a graph partitioning problem on the first level, we determine welfare-maximizing price zones and available transfer capacities for a given electricity market and analyze their impact on market outcomes. Using a generalized Benders decomposition approach developed in Grimm et al. (2019) and a problem-tailored scenario clustering for reducing the input data size, we are able to solve the model to global optimality even for large instances. We apply the approach to the German electricity market as an example to examine the impact of optimal zoning on key performance indicators such as welfare, generation mix and locations, or electricity prices. It turns out that even for a small number of price zones, an optimal configuration of zones induces a welfare level that almost approaches the first best.
Joint model of probabilistic/robust (probust) constraints applied to gas network optimization
(2017)
Optimization tasks under uncertain conditions abound in many
real-life applications. Whereas solution approaches for probabilistic constraints
are often developed in case the uncertainties can be assumed to follow a
certain probability distribution, robust approaches are usually used in case
solutions are sought that are feasible for all realizations of uncertainties within
some pre-defined uncertainty set. As many applications contain different types
of uncertainties that require robust as well as probabilistic treatments, we deal with a class of joint probabilistic/robust constraints as its appears in
optimization problems under uncertainty. Focusing on complex uncertain gas
network optimization problems, we show the relevance of this class of problems
for the task of maximizing free booked capacities in an algebraic model for a
stationary gas network. We furthermore present approaches for their solution.
Finally, we study the problem of controlling a transient system that is governed
by the wave equation. The task consists in determining controls such that a
certain robustness measure remains below some given upper bound, with high
probability.
We study gas network problems with compressors and control valves under uncertainty that can be formulated as two-stage robust optimization problems. Uncertain data are present in the physical parameters of the pipes as well as in the overall demand. We show how to exploit the special decomposable structure of the problem in order to reformulate the two-stage robust problem as a standard single-stage optimization problem. Since this structure is present in similar problems on e.g., water or direct current electricity networks, we investigate the consequences of the decomposable structure in an abstract setting: The right-hand side of the single-stage problem can be precomputed by solving a series of optimization problems and multiple elements of the right-hand side can be combined into one optimization task. In order to apply our results to gas network problems, we extend piecewise relaxations and preprocessing techniques to incorporate uncertain input data. The practical feasibility and effectiveness of our approach is demonstrated with benchmarks on realistic gas network instances. We observe large speedups due to the described aggregation method together with the developed preprocessing strategies. Furthermore, we are able to solve even comparably large gas network instances quickly for the price of slightly more conservative solutions.