the second and third constraints. The feasible solution𝑥𝑏= 0,𝑥𝑐= 5,𝑥𝑑= 10, wherethe constraints are linearly dependent, is known as adegeneratesolution. Degeneracyis a significant feature of linear programming, allowing the theoretical possibility of abreakdown in the simplex algorithm. Fortunately, such breakdown seems very rare inpractice. Degeneracy at the optimal solution indicates multiple optima.One way to proceed in this example is to arbitrarily designate one constraint as redun-dant, assuming the corresponding multiplier is zero. Arbitrarily choosing𝜆𝑚= 0 andproceeding as before, complementary slackness (𝑥𝑑>0) requires that𝐷𝑥𝑑𝐿= 3−2𝜆𝑓−𝜆𝑙= 0or𝜆𝑙= 3−2𝜆𝑓(5.106)Nonnegativity of𝜆𝑙implies that𝜆𝑓≤32.Substituting (5.106) in the second first-order condition yields𝐷𝑥𝑐𝐿= 1−2𝜆𝑓−2𝜆𝑙= 1−2𝜆𝑓−2(3−2𝜆𝑓)=−5 + 2𝜆𝑓<0 for every
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