Because cE−cI > ar gives ∂V (t,0) ∂p > 0, C 0(0) = 0 implies pFB > 0. Now we have shown pFB ∈ (0, 1), which implies (pFB(ar+ cI) + (1− p)cE) ∈ (ar+ cI , cE). Applying the condition a f(t) < ar+ cI , we get ∂V (t,p) ∂t > 0, which implies t > t. ¥ PROOF OF LEMMA 2, 3, 4. We will prove Lemma 2. Proofs of Lemma 3 and 4 are similar and thus omitted. The Kuhn-Tucker condition is −1+prf(t) ≤ 0 with eq...

This paper is concerned with the Lipschitzian behavior of the optimal set of convex semi-infinite optimization problems under continuous perturbations of the right hand side of the constraints and linear perturbations of the objective function. In this framework we provide a sufficient condition for the metric regularity of the inverse of the optimal set mapping. This condition consists of the ...

Extended Thermodynamics of dense gases is characterized by two hierarchies of field equations, which allow one to overcome some restrictions on the generality of the previous models. This idea has been introduced by Arima, Taniguchi, Ruggeri and Sugiyama. In the case of a 14-moment model, they have found the closure of the balance equations up to second order with respect to equilibrium. Here, ...

1 Proofs Proof 1 (Proof of Proposition 1) The result for u∗ is straightforward (Allen et al., 2011). We consider the problem in v: maximize v u XRv−λ||v ||1 subject to v Rv ≤ 1 & vj ≥ 0 j = 1, . . . p. (1) As this problem is convex and Slater’s condition is satisfied, the KKT conditions are necessary and sufficient for optimality. These are given by the following: RX u−λ1(p) − 2γ∗ 1 Rv∗+~γ∗ 2 =...

In this paper, we consider an optimal control problem for a linear infinite order hyperbolic system. One from the initial conditions is given by control function. Sufficient conditions for the existence of a unique solution of such hyperbolic equations with the Dirichlet boundary conditions are presented. The performance functional has the quadratic form. The time horizon T is fixed. Finally, w...

In this paper, we consider convex multiobjective optimization problems with equality and inequality constraints in real Banach space. We establish saddle point necessary sufficient Pareto optimality conditions for considered under some constraint qualifications. These results are motivated by the symmetric obtained recent article Cobos Sánchez et al. 2021 on of continuous linear operators. The ...

Determining if a solution is optimal or near optimal is fundamental in optimization theory, algorithms, and computation. For instance, Karush-Kuhn-Tucker conditions provide necessary and sufficient optimality conditions for certain classes of problems, and bounds on optimality gaps are frequently used as part of optimization algorithms. Such bounds are obtained through Lagrangian, integrality, ...

In this paper, we study necessary optimality conditions for nonsmooth mathematical programs with equilibrium constraints. We first show that, unlike the smooth case, the Mathematical Program with Equilibrium Constraints Linear Independent Constraint Qualification is not a constraint qualification for the strong stationary condition when the objective function is nonsmooth. We argue that the str...

Let f be an integrable function on an infinite measure space (S,S , π). We show that if a regenerative sequence {Xn}n≥0 with canonical measure π could be generated then a consistent estimator of λ ≡ ∫ S fdπ can be produced. We further show that under appropriate second moment conditions, a confidence interval for λ can also be derived. This is illustrated with estimating countable sums and inte...

In this paper, a model of an optimal control problem with chance constraints is introduced. The parametersof the constraints are fuzzy, random or fuzzy random variables. Todefuzzify the constraints, we consider possibility levels.  Bychance-constrained programming the chance constraints are converted to crisp constraints which are neither fuzzy nor stochastic and then the resulting classical op...