By Aharon Ben-Tal, Laurent El Ghaoui, Arkadi Nemirovski

Robust optimization remains to be a comparatively new method of optimization difficulties plagued by uncertainty, however it has already proved so helpful in genuine functions that it's tough to take on such difficulties this present day with no contemplating this robust method. Written by way of the relevant builders of sturdy optimization, and describing the most achievements of a decade of study, this can be the 1st booklet to supply a entire and updated account of the subject.

Robust optimization is designed to satisfy a few significant demanding situations linked to uncertainty-affected optimization difficulties: to function less than loss of complete details at the nature of uncertainty; to version the matter in a sort that may be solved successfully; and to supply promises concerning the functionality of the solution.

The publication starts off with a comparatively uncomplicated therapy of doubtful linear programming, continuing with a deep research of the interconnections among the development of acceptable uncertainty units and the classical probability constraints (probabilistic) procedure. It then develops the strong optimization idea for doubtful conic quadratic and semidefinite optimization difficulties and dynamic (multistage) difficulties. the speculation is supported by way of a variety of examples and computational illustrations.

An crucial ebook for someone engaged on optimization and determination making lower than uncertainty, strong Optimization additionally makes a terrific graduate textbook at the subject.

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3. Tractability of the RC of an uncertain LO problem with a tractable uncertainty set was established in the very first papers on convex RO. 5 are taken from [5]. 1 HOW TO SPECIFY AN UNCERTAINTY SET The question posed in the title of this section goes beyond general-type theoretical considerations — this is mainly a modeling issue that should be resolved on the basis of application-driven considerations. There is however a special case where this question makes sense and can, to some extent, be answered — this is the case where our goal is not to build an uncertainty model “from scratch,” but rather to translate an already existing uncertainty model, namely, a stochastic one, to the language of “uncertain-but-bounded” perturbation sets and the associated robust counterparts.

P , so that Z is nothing but the set L UNCERTAIN LINEAR OPTIMIZATION PROBLEMS AND THEIR ROBUST COUNTERPARTS 25 P × ... × P , where P = Conv(P ). 4. 1. Consider an uncertain LO problem with instances min cT x : Ax ≤ b [A : m × n] x and with simple interval uncertainty: U = {(c, A, b) : |cj − cnj | ≤ σj , |Aij − Anij | ≤ αij , |bi − bni | ≤ βi ∀i, j} (n marks the nominal data). Reduce the RC of the problem to an LO problem with m constraints (not counting the sign constraints on the variables) and 2n nonnegative variables.

A. , S}. , S, where is the cone dual to K . B. 4. 6) in two particular cases. While at a first glance no natural “uncertainty models” lead to the “strange” perturbation sets we are about to consider, it will become clear later that these sets are of significant importance — they allow one to model random uncertainty. 7. 15) =1 where σ > 0 and Ω > 0 are given parameters. , L, τ1 [⇔ [η1 ; τ1 ] ∈ K1∗ ], τ2 [⇔ [η2 ; τ2 ] ∈ K2∗ ]. We can eliminate from this system the variables τ1 , τ2 — for every feasible solution to the system, we have τ1 ≥ τ¯1 ≡ η1 1 , τ2 ≥ τ¯2 ≡ η2 2 , and the solution obtained when replacing τ1 , τ2 with τ¯1 , τ¯2 still is feasible.

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