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Convex Optimization by Lieven Vandenberghe, Stephen Boyd

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4.7.2  Optimal points and values

We first consider a special case, in which the meaning of the vector optimization problem is clear. Consider the set of objective values of feasible points,

which is called the set of achievable objective values. If this set has a minimum element (see §2.4.2), i.e., there is a feasible x such that f0(x) f0(y) for all feasible y, then we say x is optimal for the problem (4.56), and refer to f0(x) as the optimal value of the problem. (When a vector optimization problem has an optimal value, it is unique.) If

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