arXiv subjects

Markus Gabl

Publications and source records attributed to Markus Gabl.

5 recordsLinked to original sources

arXiv2025

Conic optimization techniques yield sufficient conditions for set-completely positive matrix completion under arrowhead specification pattern

Matrix completion results deal with the question of when a partially specified symmetric matrix can be completed to a member of certain matrix cones. Results from positive semidefinite matrix completion and completely positive matrix completion have been successfully applied in optimization to greatly reduce the number of variables in conic optimization problems in the space of symmetric matrices. In this text, we go the other direction and show that we can use tools from conic optimization (more precisely: from copositive optimization) to establish a new completion result that complements the existing literature in two regards: firstly, we consider set-completely positive matrix completion, which generalizes completion with respect to the traditional completely positive matrix cone. Secondly, we consider a specification pattern that is not in the scope of classical results for completely positive matrix completion. Namely, we consider arrow-head specification patterns where the width is equal to one. Our theory is applied to a class of quadratic optimization problems.

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arXiv2024

Finding quadratic underestimators for optimal value functions of nonconvex all-quadratic problems via copositive optimization

Modeling parts of an optimization problem as an optimal value function that depends on a top-level decision variable is a regular occurrence in optimization and an essential ingredient for methods such as Benders Decomposition. It often allows for the disentanglement of computational complexity and exploitation of special structures in the lower-level problem that define the optimal value functions. If this problem is convex, duality theory can be used to build piecewise affine models of the optimal value function over which the top-level problem can be optimized efficiently. In this text, we are interested in the optimal value function of an all-quadratic problem (also called quadratically constrained quadratic problem, QCQP) which is not necessarily convex, so that duality theory can not be applied without introducing a generally unquantifiable relaxation error. This issue can be bypassed by employing copositive reformulations of the underlying QCQP. We investigate two ways to parametrize these by the top-level variable. The first one leads to a copositive characterization of an underestimator that is sandwiched between the convex envelope of the optimal value function and that envelope's lower-semicontinuous hull. The dual of that characterization allows us to derive affine underestimators. The second parametrization yields an alternative characterization of the optimal value function itself, which other than the original version has an exact dual counterpart. From the latter, we can derive convex and nonconvex quadratic underestimators of the optimal value function. In fact, we can show that any quadratic underestimator is associated with a dual feasible solution in a certain sense.

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arXiv2024

Concave tents: a new tool for constructing concave reformulations of a large class of nonconvex optimization problems

Optimizing a nonlinear function over nonconvex sets is challenging since solving convex relaxations may lead to substantial relaxation gaps and infeasible solutions that must be "rounded" to feasible ones, often with uncontrollable losses in objective function performance. For this reason, these convex hulls are especially useful if the objective function is linear or even concave, since concave optimization is invariant to taking the convex hull of the feasible set. We propose the notion of concave tents, which are concave approximations of the original objective function that agree with this objective function on the feasible set, and allow for concave reformulations of the problem. Concave tents, therefore, are special cases of concave extensions. In this text, we derive such concave tents for a large class of objective functions as the optimal value functions of conic optimization problems. Hence, evaluating our concave tents requires solving a conic problem. We can find supergradients by solving the conic dual problem, so that differentiation is of the same complexity as evaluation. For feasible sets that are contained in the extreme points of their convex hull, we construct these concave tents in the original space of variables. For general feasible sets, we propose a double lifting strategy, where the original optimization problem is lifted into a higher-dimensional space in which the concave tent can be constructed easily. We investigate the relation of the so-constructed concave tents to concave envelopes and naive concave tents based on concave quadratic updates. Based on these ideas, we propose a primal heuristic for a class of robust discrete quadratic optimization problems that can be used instead of classical rounding techniques. Numerical experiments suggest that our techniques can be beneficial as an upper-bounding procedure in a branch-and-bound solution scheme.

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arXiv2021

Sparse Conic Reformulation of Structured QCQPs based on Copositive Optimization with Applications in Stochastic Optimization

In an effort to develop an alternative approach to traditional sparse reformulations, we will provide a new type of convex reformulation of a large class of stochastic quadratically constrained quadratic optimization problems that is similar to Burer's reformulation, but lifts the variables into a comparatively lower dimensional space. The reformulation rests on a generalization of the set-completely positive matrix cone. This cone can then be approximated via inner and outer approximations in order to obtain upper and lower bounds, which potentially close the optimality gap, and hence can give a certificate of exactness for these sparse reformulations outside of traditional, known sufficient conditions. Finally, we provide some numerical experiments, where we asses the quality of the inner and outer approximations, thereby showing that the approximations may indeed close the optimality gap in interesting cases.

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arXiv2020

Uncertainty Preferences in Robust Mixed-Integer Linear Optimization with Endogenous Uncertainty

In robust optimization one seeks to make a decision under uncertainty, where the goal is to find the solution with the best worst-case performance. The set of possible realizations of the uncertain data is described by a so-called uncertainty set. In many scenarios, a decision maker may influence the uncertainty regime she is facing, for example, by investing in market research, or in machines which work with higher precision. Recently, this situation was addressed in the literature by introducing decision dependent uncertainty sets (endogenous uncertainty), i.e., uncertainty sets whose structure depends on (typically discrete) decision variables. In this way, one can model the trade-off between reducing the cost of robustness versus the cost of the investment necessary for influencing the uncertainty. However, there is another trade-off to be made here. With different uncertainty regimes, not only do the worst-case optimal solutions vary, but also other aspects of that solutions such as max-regret, best-case performance or predictability of the performance. A decision maker may still be interested in having a performance guarantee, but at the same time be willing to forgo superior worst-case performance if those other aspects can be enhanced by switching to a suitable uncertainty regime. We introduce the notion of uncertainty preference in order to capture such stances. We present three ways to formalize uncertainty preferences and study the resulting mathematical models. The goal is to have reformulations/approximations of these models which can be solved with standard methods. The workhorse is mixed-integer linear and conic optimization. We apply our framework to the uncertain shortest path problem and conduct numerical experiments for the resulting models. We can demonstrate that our models can be handled very well by standard mixed-integer linear solvers.

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