Γ-robust linear complementarity problems

Krebs, Vanessa; Schmidt, Martin

doi:10.1080/10556788.2020.1825708

Cited by 9 publications

(15 citation statements)

References 31 publications

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“…This means, there are at most

{normalΓ}_{M} \in false{1, \dots, n^{2} false}

many values in

M (u_{1})

and

{normalΓ}_{q} \in false{1, \dots, n false} = : false[n false]

many values in

q (u_{2})

, which are uncertain and can thus realize in a worst‐case way. In the paper by Krebs and Schmidt (2020), we already considered Problem (4) for uncertainty realizing in q or M and for box‐ and ℓ 1 ‐norm uncertainty. Here, we consider the case of ellipsoidal uncertainty sets.…”

Section: Problem Statementmentioning

confidence: 99%

“…Thus, alternative robustness concepts have been developed starting with the Γ‐approach introduced in Bertsimas and Sim (2004) that we also study in this paper in the context of uncertain LCPs. The Γ‐approach for uncertain LCPs has also been studied in the recent paper by Krebs and Schmidt (2020), where ℓ 1 ‐ and box‐uncertainty sets have been considered. The present paper is an extension of the latter work and studies Γ‐robustified LCPs with ellipsoidal uncertainty sets.…”

Section: Introductionmentioning

confidence: 99%

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Γ‐robust linear complementarity problems with ellipsoidal uncertainty sets

Krebs

Müller

Schmidt

2021

Int Trans Operational Res

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We study uncertain linear complementarity problems (LCPs), that is, problems in which the LCP vector q or the LCP matrix M may contain uncertain parameters. To this end, we use the concept of Γ‐robust optimization applied to the gap function formulation of the LCP. Thus, this work builds upon Krebs and Schmidt (2020). There, we studied Γ‐robustified LCPs for ℓ1‐ and box‐uncertainty sets, whereas we now focus on ellipsoidal uncertainty sets. For uncertainty in q or M, we derive conditions for the tractability of the robust counterparts. For these counterparts, we also give conditions for the existence and uniqueness of their solutions. Finally, a case study for the uncertain traffic equilibrium problem is considered, which illustrates the effects of the values of Γ on the feasibility and quality of the respective robustified solutions.

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“…This means, there are at most

{normalΓ}_{M} \in false{1, \dots, n^{2} false}

many values in

M (u_{1})

and

{normalΓ}_{q} \in false{1, \dots, n false} = : false[n false]

many values in

q (u_{2})

Section: Problem Statementmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Γ‐robust linear complementarity problems with ellipsoidal uncertainty sets

Krebs

Müller

Schmidt

2021

Int Trans Operational Res

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“…Using the techniques as in, e.g., [39,43], we obtain the following equivalent reformulation of the robust counterpart (9). Theorem 3.1 The -robust counterpart (9) of the welfare maximization problem (6) is equivalent to…”

Section: An Equivalent Variational Inequalitymentioning

confidence: 99%

“…Again, using the techniques as in, e.g., [39,43], we obtain the following reformulation of the robust counterpart (13).…”

Section: An Equivalent Variational Inequalitymentioning

confidence: 99%

“…However, the stochastic approach to LCPs is rather mature (see, e.g., [10][11][12]44]) compared to the field of robust LCPs or robust market equilibrium problems, which are still in their infancies. The only studies we are aware of are [7,40,43,[54][55][56] on robust LCPs, [39,45] on robust market equilibrium modeling, and the very recent and related paper [18] on robust bidding strategies in auctions. In this paper, we contribute to the second of the three mentioned fields and consider robustified market equilibrium models with transmission and generation investments.…”

Section: Introductionmentioning

confidence: 99%

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$$\Gamma $$-Robust electricity market equilibrium models with transmission and generation investments

2021

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We consider uncertain robust electricity market equilibrium problems including transmission and generation investments. Electricity market equilibrium modeling has a long tradition but is, in most of the cases, applied in a deterministic setting in which all data of the model are known. Whereas there exist some literature on stochastic equilibrium problems, the field of robust equilibrium models is still in its infancy. We contribute to this new field of research by considering $$\Gamma $$ Γ -robust electricity market equilibrium models on lossless DC networks with transmission and generation investments. We state the nominal market equilibrium problem as a mixed complementarity problem as well as its variational inequality and welfare optimization counterparts. For the latter, we then derive a $$\Gamma $$ Γ -robust formulation and show that it is indeed the counterpart of a market equilibrium problem with robustified player problems. Finally, we present two case studies to gain insights into the general effects of robustification on electricity market models. In particular, our case studies reveal that the transmission system operator tends to act more risk-neutral in the robust setting, whereas generating firms clearly behave more risk-averse.

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