Robust Quadratic Programming with Mixed-Integer Uncertainty

Mittal, Areesh; Gokalp, Can; Hanasusanto, Grani A.

doi:10.1287/ijoc.2019.0901

Cited by 9 publications

(9 citation statements)

References 43 publications

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“…There has been previous work on developing exact copositive programming reformulations for otherwise difficult problems, and using those reformulations to generate tractable approximations [8,11,13,31,19,32,34]. Our results add to this literature by demonstrating the ability of generalized copositive programs to exactly model the MVEP.…”

Section: Introductionmentioning

confidence: 55%

Finding Minimum Volume Circumscribing Ellipsoids Using Generalized Copositive Programming

Mittal¹,

Hanasusanto²

2018

Preprint

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We study the problem of finding the Löwner-John ellipsoid, i.e., an ellipsoid with minimum volume that contains a given convex set. We reformulate the problem as a generalized copositive program, and use that reformulation to derive tractable semidefinite programming approximations for instances where the set is defined by affine and quadratic inequalities. We prove that, when the underlying set is a polytope, our method never provides an ellipsoid of higher volume than the one obtained by scaling the maximum volume inscribed ellipsoid. We empirically demonstrate that our proposed method generates high-quality solutions faster than solving the problem to optimality. Furthermore, we outperform the existing approximation schemes in terms of solution time and quality. We present applications of our method to obtain piecewise-linear decision rule approximations for dynamic distributionally robust problems with random recourse, and to generate ellipsoidal approximations for the set of reachable states in a linear dynamical system when the set of allowed controls is a polytope.

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Section: Introductionmentioning

confidence: 55%

Finding Minimum Volume Circumscribing Ellipsoids Using Generalized Copositive Programming

Mittal¹,

Hanasusanto²

2018

Preprint

Self Cite

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“…Many results in recent literature have harnessed this general strategy in order to cope with cases where quadratic terms in the uncertainty vector appear (see e.g. Mittal et al 2019;Xu and Hanasusanto 2019). We want to highlight that the critical ingredients for the above strategy are:…”

Section: Convex Reformulations Of Qcqps and Robust Optimizationmentioning

confidence: 99%

“…The theorem highlights the fact, that in absence of dual attainability the feasible set is shrinking, and locates the points that are lost at the boundary of the setcopositive cone we sought to characterize. The reformulation results in Burer (2009), Eichfelder and Povh (2013) have been used in Mittal et al (2019), Xu and Hanasusanto (2019) for the sake of reformulating semi-infinite constraints with quadratic dependency on the uncertainty vector. We want to point out that in these papers the problem of dual attainability was not sufficiently considered when taking uncertain constraints into account.…”

Section: Remarkmentioning

confidence: 99%

“…The argument for the linearization of the constraint H = (W, w) T (W, w) in the form M (E r , 2(W, w), H) ∈ S n 1 +r +1 + is given in the following lemma, which is a straightforward generalization of (Mittal et al 2019, Lemma 4). It is slightly more general than needed for Theorem 19 as it might be useful in other contexts such as the ones discussed in Mittal et al (2019).…”

Section: Theorem 19mentioning

confidence: 99%

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Interplay of non-convex quadratically constrained problems with adjustable robust optimization

Bomze

Gabl

2020

Math Meth Oper Res

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In this paper we explore convex reformulation strategies for non-convex quadratically constrained optimization problems (QCQPs). First we investigate such reformulations using Pataki’s rank theorem iteratively. We show that the result can be used in conjunction with conic optimization duality in order to obtain a geometric condition for the S-procedure to be exact. Based upon known results on the S-procedure, this approach allows for some insight into the geometry of the joint numerical range of the quadratic forms. Then we investigate a reformulation strategy introduced in recent literature for bilinear optimization problems which is based on adjustable robust optimization theory. We show that, via a similar strategy, one can leverage exact reformulation results of QCQPs in order to derive lower bounds for more complicated quadratic optimization problems. Finally, we investigate the use of reformulation strategies in order to derive characterizations of set-copositive matrix cones. Empirical evidence based upon first numerical experiments shows encouraging results.

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“…Consequently, a quantitative decision could be made by referring to the optimal solution obtained from solving the quadratic programming problem, especially with the fuzzy parameters [8,9,10]. Also, the computational techniques for solving the quadratic programming problem under the probabilistic environment [11] and the related robust solution [12] are actively studied.…”

Section: Introductionmentioning

confidence: 99%

Unconstrained Quadratic Programming Problem with Uncertain Parameters

Kek¹,

Lim²,

Ooi³

2021

GJRE

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In this paper, an unconstrained quadratic programming problem with uncertain parameters is discussed. For this purpose, the basic idea of optimizing the unconstrained quadratic programming problem is introduced. The solution method of solving linear equations could be applied to obtain the optimal solution for this kind of problem. Later, the theoretical work on the optimization of the unconstrained quadratic programming problem is presented. By this, the model parameters, which are unknown values, are considered. In this uncertain situation, it is assumed that these parameters are normally distributed; then, the simulation on these uncertain parameters are performed, so the quadratic programming problem without constraints could be solved iteratively by using the gradient-based optimization approach. For illustration, an example of this problem is studied. The computation procedure is expressed, and the result obtained shows the optimal solution in the uncertain environment. In conclusion, the unconstrained quadratic programming problem, which has uncertain parameters, could be solved successfully.

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Robust Quadratic Programming with Mixed-Integer Uncertainty

Cited by 9 publications

References 43 publications

Finding Minimum Volume Circumscribing Ellipsoids Using Generalized Copositive Programming

Finding Minimum Volume Circumscribing Ellipsoids Using Generalized Copositive Programming

Interplay of non-convex quadratically constrained problems with adjustable robust optimization

Unconstrained Quadratic Programming Problem with Uncertain Parameters

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