Perspective envelopes for bilinear functions

Hijazi, Hassan

doi:10.1063/1.5089984

Cited by 5 publications

(3 citation statements)

References 38 publications

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“…13), with respect to the variables set Ξ ul Y Ξ r Y Ξ du . Note that equations ( 1)-( 3) and (B) are from the Part I paper, where (1) denote the upper-level constraints, constraints (2) lower-level primal constraints (primal feasibility KKT conditions), (3) are reformulated lowerlevel SOC constraints, i.e. reformulated constraints (2), while (B) are lower-level dual constraints, i.e.…”

Section: A Primal-dual Counterpart 1) Primal-dual (Pd)mentioning

confidence: 99%

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Solving Bilevel AC OPF Problems by Smoothing the Complementary Conditions -- Part II: Solution Techniques and Case Study

Sepetanc¹,

Pandžić²,

Capuder³

2022

Preprint

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This is a second part of the research on AC optimal power flow being used in the lower level of the bilevel strategic bidding or investment models. As an example of a suitable upper-level problem, we observe a strategic bidding of energy storage and propose a novel formulation based on the smoothing technique.After presenting the idea and scope of our work, as well as the model itself and the solution algorithm in the companion paper (Part I), this paper presents a number of existing solution techniques and the proposed one based on smoothing the complementary conditions. The superiority of the proposed algorithm and smoothing techniques is demonstrated in terms of accuracy and computational tractability over multiple transmission networks of different sizes and different OPF models. The results indicate that the proposed approach outperforms all other options in both metrics by a significant margin. This is especially noticeable in the metric of accuracy where out of total 422 optimizations over 9 meshed networks the greatest AC OPF error is 0.023% that is further reduced to 3.3e-4% in the second iteration of our algorithm.

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Section: A Primal-dual Counterpart 1) Primal-dual (Pd)mentioning

confidence: 99%

“…McCormick envelopes [3] relax a bilinear term into a planebounded region. The technique requires an assumption on the bounds of the electricity price, i.e.…”

Section: Mccormick Envelopes (Mc)mentioning

confidence: 99%

Solving Bilevel AC OPF Problems by Smoothing the Complementary Conditions -- Part II: Solution Techniques and Case Study

Sepetanc¹,

Pandžić²,

Capuder³

2022

Preprint

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“…He characterized the convex and concave envelope of x i x j over a triangular domain and used it to improve upon (4). Based on perspective functions, Hijazi [26] derived a closed formula for the convex and concave envelope on a polytope of the form…”

Section: Convexification Of Bilinear Termsmentioning

confidence: 99%

Using Two-Dimensional Projections for Stronger Separation and Propagation of Bilinear Terms

Müller¹,

Serrano²,

Gleixner³

2020

SIAM J. Optim.

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One of the most fundamental ingredients in mixed-integer nonlinear programming solvers is the well-known McCormick relaxation for a product of two variables x and y over a boxconstrained domain. The starting point of this paper is the fact that the convex hull of the graph of xy can be much tighter when computed over a strict, non-rectangular subset of the box. In order to exploit this in practice, we propose to compute valid linear inequalities for the projection of the feasible region onto the x-y-space by solving a sequence of linear programs akin to optimization-based bound tightening. These valid inequalities allow us to employ results from the literature to strengthen the classical McCormick relaxation. As a consequence, we obtain a stronger convexification procedure that exploits problem structure and can benefit from supplementary information obtained during the branchand bound algorithm such as an objective cutoff. We complement this by a new bound tightening procedure that efficiently computes the best possible bounds for x, y, and xy over the available projections. Our computational evaluation using the academic solver SCIP exhibit that the proposed methods are applicable to a large portion of the public test library MINLPLib and help to improve performance significantly.

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Tight Convex Relaxations for the Expansion Planning Problem

Lenz

Serrano

2022

J Optim Theory Appl

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Secure energy transport is considered as highly relevant for the basic infrastructure of nowadays society and economy. To satisfy increasing demands and to handle more diverse transport situations, operators of energy networks regularly expand the capacity of their network by building new network elements, known as the expansion planning problem. A key constraint function in expansion planning problems is a nonlinear and nonconvex potential loss function. In order to improve the algorithmic performance of state-of-the-art MINLP solvers, this paper presents an algebraic description for the convex envelope of this function. Through a thorough computational study, we show that this tighter relaxation tremendously improves the performance of the MINLP solver SCIP on a large test set of practically relevant instances for the expansion planning problem. In particular, the results show that our achievements lead to an improvement of the solver performance for a development version by up to 58%.

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Perspective envelopes for bilinear functions

Cited by 5 publications

References 38 publications

Solving Bilevel AC OPF Problems by Smoothing the Complementary Conditions -- Part II: Solution Techniques and Case Study

Solving Bilevel AC OPF Problems by Smoothing the Complementary Conditions -- Part II: Solution Techniques and Case Study

Using Two-Dimensional Projections for Stronger Separation and Propagation of Bilinear Terms

Tight Convex Relaxations for the Expansion Planning Problem

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