Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York (1982)  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(2.1) For the purposes of a pattern search augmented Lagrangian approach, which assumes no explicit knowledge of derivative information, one appears to have no choice other than some variant of the Hestenes Powell multiplier update. All other multiplier update formulae (such as those discussed in [1, 32]) require information about derivatives. The projection onto the convex set B = will be denoted by P ; it is defined component wise by (P [x] i = # # # # i if x i u i if x i u i x i otherwise. Given x B and a vector v, we define P (x, v) x P [x v] Unless ....

....union of J 1 and any subset of J 2 . The next result from [6] which also holds for the augmented Lagrangian pattern search algorithm, is Lemma 5.1. This result relates the convergence of the iterates to the error in the multipliers, a relationship characteristic of augmented Lagrangian methods [1, 32]. Again, the proof in [6] holds for the pattern search variant because of Proposition 5.1. Lemma 5.3. Suppose that (AS1) holds. Let #B, K, be a subsequence which converges to the Karush Kuhn Tucker point x # for which (AS2) AS4) and (AS5) hold, and let # # be the corresponding vector of ....

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D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, NY, 1982.

....coincide with the linearized cone. The next issue that deserves to be discussed is reformulating inequality constraints as equalities, with the aim of subsequently using results available for the latter. This technique is known to be useful for regular inequality constrained problems; e.g. see [9]. Analogously, one might try to apply known optimality conditions for (irregular) equality constrained problems to reformulations of irregular inequality constraints. For example, the theory of 2 regularity [29, 4, 6, 5, 16, 1, 13, 20, 17, 15] o#ers optimality conditions for the case in which ....

D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, 1982.

....have positive values, i.e. 0 n a for 1 , 1 = N n . Addressing point 1 consists of solving the linear system of equations of eqn. 10) with the slope constraints mentioned. This is essentially the solution of a quadratic penalty function (due to the least squares) with slope constraints [12] that is typically solved numerically. We have used the built in Matlab function lsqnonneg from the optimization toolbox to obtain the solution. The second point is of practical concern and is usually noticed when the two images to register in range are both under or over exposed. In this case, ....

D.P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, New York: Academic Press, 1982.

....for example [8, 21, 22] Further, for simplicity of notation, we consider the case when i , i are independent of i; the extension onto the general case is quite straightforward. Augmented Lagrangian approach Methods of multipliers, involving nonquadratic augmented Lagrangians [10, 6, 15, 3, 19, 20, 4, 7, 11] successfully compete with the interior point and other methods in non linear and semidefinite programming. They are especially ecient when a very high accuracy of solution is required. This success is partially explained by the fact, that due to iterative update of multipliers, the penalty ....

Bertsekas, D.P. (1982). Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York.

....determined by the particular choice of the functions and . Some additional restrictions on updating of multipliers are introduced [2] in order to stabilize iterative process. Well known penalty functions, which are used in nonquadratic Augmented Lagrangian methods, are exponential penalty [3] and shifted logarithmic barrier [6] We introduce and prefer to use in our computations new quadratic logarithmic penalty function (t) a 2 t bt c t d log(t e) f t where 1 1 is a parameter fixing the joint point. The coefficients a; b; c, d; e; f are uniquely ....

Bertsekas, D. P. (1982). Constrained Optimization and Lagrange Multiplier Method, Academic Press, NY.

....For instance, two important applications are min optimization and vertical linear complementarity (VLCP) problems. In min optimization the smooth approximation is directly used to replace the max function, and so that the resulting model becomes a regular nonlinear programming problem [2]. On the other hand, in VLCP problems the system of equalities and nonnegativity constraints is equivalently modelled as a system of equalities composed of high dimensional max functions. After replacing the max functions with their approximations, a system of nonlinear inequalities are formed. ....

....most frequently used approximation in the literature: 4.1) g(x) log . This function shows the following convergence property lim t#0 tg(t 1 x) lim t#0 t log = max x 1 , Therefore, 4. 1) is successfully utilized in solving both min optimization and VLCP problems [2, 10]. Notice that an overflow easily occurs when the exponential function in (4.1) is computed with a very large argument. A well known trick to handle this potential problem is introducing a constant z and then computing (4.2) g z (x; t) t log x i z z. In order to apply the ....

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Bertsekas, D.P. Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York, 1982.

.... closed convex set, this method reduces to a projection method proposed by Sibony [23] for monotone variational inequalities and, in the further case where B is the gradient of a dioeerentiable convex function, it amounts to a gradient projection method of Goldstein and of Levintin and Polyak [5]. This method was largely analyzed by Mercier [14] and Gabay [9] They namely showed that if B is cocoercive with modulus fl 0, then the iterates x k converge weakly to a solution on condition that k is constant and less than 2fl. The case where k is noconstant was dealt with among others in ....

D. P. Bertsekas, Constrained optimization and lagrange multiplier methods, Academic Press, New York, 1982.

.... minimum may be achieved due to the non convexity behavior of this optimization problem [7] Besides, neither of them considered the individual transmit powers constraints (8) The gradient technique presented in [5] was based on the Lagrange multipliers method and the quadratic penalty function [8]. This technique may have some convergence problems related to the speed of convergence and the convergence itself [5] 8] Besides, the gradient requires that the constraints are differentiable, limiting its application. In the simulation results section, we compare a gradient based technique ....

.... the individual transmit powers constraints (8) The gradient technique presented in [5] was based on the Lagrange multipliers method and the quadratic penalty function [8] This technique may have some convergence problems related to the speed of convergence and the convergence itself [5] [8]. Besides, the gradient requires that the constraints are differentiable, limiting its application. In the simulation results section, we compare a gradient based technique with the proposed algorithm in this paper, based on Simulated Annealing (SA) In this paper we propose the application of SA ....

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Dimitri P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Computer Science and Applied Mathematics, Academic Press, 1982.

....# j . Moreover, we have by (A5) that (Y (x # j s # j c(x j s # j )# # #(Y E#A ##c E#A (x # j s # j )# ) Hereby, we have used that L y and that c (x # j s # j ) # j A # j s # j O(#s # j ) O(#s # j ) by (37) Thus, we obtain with some # [0, 1] #T j (# xx #(x # j #s # j , y # j ) B # j )s # j c(x # j s # j )# ) 38) On the other hand, using the step decomposition (12) and the obvious inequality #n j s #t j 2(#s #n j we have by (15) and (14) after a possible increase of K for all j #q ....

D. P. BERTSEKAS, Constrained optimization and Lagrange multiplier methods, Computer Science and Applied Mathematics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982.

....ellipsoids. Two special cases of the result presented above occur when no estimate for one of the bounds ei or E is available and when E becomes infinite. In the first case, the prior knowledge defines all but one of the ellipsoids Q, or Q. Then the constrained least squares (CLS) approach [3, 8, 14] results in a restored image which is on the surface of the ellipsoids which are defined at the point of minimum distance from the center of the ellipsoid which is not defined. Two of these solutions, denoted by xL, when E is unknown, and : when e is known, are shown in Fig. I. As can be seen ....

D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, 1982.

....jj denotes Euclidean norm, and jj jj is the penalty term to ensure that the optimization problem is held at the condition of local convexity assumption: r XX L 0. We use the augmented Lagrangian function in this paper because it gives wider applicability and provides better stability [6]. For discrete problems, the changes in the augmented Lagrangian function can be de ned as XL(X; to achieve the saddle point in the discrete variable space. The iterative equations to solve the problem in eq. 2) are given as follows: X(k 1) X(k) XL(X(k) k) k) k 1) k) ....

Dimitri P. Bertsekas. Constrained optimization and Lagrange multiplier methods. New York : Academic Press, 1982.

....values. Let I d be the set of indices of discrete variables, and I c be those of continuous variables. Then I d , n , and I d #. Variable space X is infinite because of continuous variables. 1. 2 Basic Concepts To characterize the solutions sought, we introduce some basic concepts [31, 125, 145, 69] on neighborhoods, feasible solutions, and constrained local and global minima here. Definition 1.1 (x) the neighborhood of point x in variable space X, is a set of points X such that x # (x # ) Neighborhood (x) has di#erent meanings for the three types of constrained NLPs ....

....Normally, one may choose dn (x) to include nearby discrete points to x so that neighborhood carries its original meaning. However, one may also choose the neighborhood to contain far away points. For a continuous problem, neighborhood cn (x) is well defined and application independent [31, 125, 106]. It includes those points that are su#ciently close to x, i.e. cn (x) is a set of points x # such that x x # for some small # 0. For a mixed integer problem, x) can be defined as a joint neighborhood of its discrete subspace x i (i I d ) and its continuous subspace x j (j ....

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D. P. Bertsekas. Constrained Optimization and Lagrange Multiplier Methods. Academic Press, 1982.

....Lagrange Multipliers for Solving Discrete Constrained NLPs In this section, we briefly overview a new theory of discrete constrained optimization using Lagrange multipliers [163, 169] developed in our research group. In contrast to Lagrangian methods that work only for continuous constrained NLPs [45, 109], our new theory was derived for solving discrete constrained NLPs and can be extended to solve both continuous and mixed integer NLPs. More importantly, its first order necessary and su#cient condition on CLM dn provides a strong theoretic foundation for developing global optimization methods for ....

....solving the penalty function by unconstrained methods. A typical penalty formulation is as follows: eval(x) f(x) w i i (x) 2.3) where f(x) is the objective function, and w i is the i weight coe#cient to be determined. A simple solution is to use a static penalty formulation [45, 109] that sets w i to be a large static positive value. This way, a local minimum of eval(x) is a CLM dn , and a global minimum of eval(x) is a CGM dn . However, if the w i s are too large, they will cause the search space to be very rugged. Consequently, feasible solutions are di#cult to be located ....

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D. P. Bertsekas. Constrained Optimization and Lagrange Multiplier Methods. Academic Press, 1982.

....formulation is as follows: eval(x) f(x) p(x) 2.1) where f(x) is the objective function and p(x) is the penalty term. A widely used penalty term is: p(x) w i i (x) 2.2) where w i are weight coe#cients to be determined. A simple solution is to use a static penalty formulation [22, 128] that sets w i to be static large positive values. This way, a local minimum of eval(x) is a constrained local minimum (CLM dn ) and a global minimum of eval(x) is a constrained global minimum (CGM dn ) However, if the w i s are too large, they will cause the search space to be very rugged. ....

....of unconstrained subproblems with increasing penalties, dynamic penalty methods employ the solution in a previous subproblem as a starting point for the next subproblem. Dynamic penalty methods have asymptotic convergence if each unconstrained subproblem in the sequence is solved optimally [22, 128]. Optimality in each subproblem is, however, di#cult to achieve in practice, given only finite amount of time to solve each subproblem, leading to suboptimal solutions when the result in one subproblem is not optimal. Moreover, the solutions to intermediate subproblems may not be related to the ....

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D. P. Bertsekas. Constrained Optimization and Lagrange Multiplier Methods. Academic Press, 1982.

....is as follows: cost(x) f(x) w i i (x) 2.3) where f(x) is the objective function, and w i is the i th weight coe#cient. The problem is how to determine the w # i s. A simple solution is to set w i to be a large constant positive value, resulting in a staticpenalty formulation [38, 106]. If the w i s are large enough, a local minimum of cost(x) is most likely a CLM dn , and a global minimum of cost(x) is a CGM dn . However, if the w i s are too large, the search space is very rugged. Consequently, it is di#cult to locate feasible solutions using local search methods because ....

....(1.1) into a sequence of unconstrained subproblems with increasing penalties, and use the solution of a previous subproblem as a starting point for the next subproblem. Dynamic penalty methods have asymptotic convergence if each unconstrained subproblem in the sequence can be solved optimally [38, 106]. However, it is di#cult to achieve optimality in each subproblem in practice, given finite time to solve each subproblem. This results in suboptimal solutions when the solutions in at least one subproblem are not optimal. Moreover, the solutions to the first few subproblems may not be related to ....

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D. P. Bertsekas. Constrained Optimization and Lagrange Multiplier Methods. Academic Press, 1982.

....and then solves it by using existing unconstrained minimization methods. Many heuristics developed to handle constraints [7] are normally problem dependent, have di#culties in finding feasible regions or in maintaining feasibility, and get stuck easily in local minima. Static penalty formulations [3, 6] transform (1) into an unconstrained problem, min x L # (x, #) f(x) i (x) #m j max where # 0, and penalty # = 1 , # 2 , #m k is fixed and chosen to be large enough so that L # (x # , #) L # (x, #) #x # opt and x # X opt . 3) Based on (3) an ....

....# = 1, 2, K, where 0 #(#) #(# 1) and #(K) #. Here # # # i# # i # # i for every i = 1, 2, m, and # # # i# # # # and there exists at least one i such that # i # # i . Dynamic penalty methods are asymptotically convergent if, for every #(#) 4) is solved optimally [3, 6]. The requirement of obtaining an unconstrained global minimum of (4) in every stage is, however, di#cult to achieve in practice, given only finite amount of time in each stage. If the result in one stage is not a global minimum, then the process cannot be guaranteed to find a constrained global ....

D. P. Bertsekas. Constrained Optimization and Lagrange Multiplier Methods. Academic Press, 1982.

....the RMVB is guaranteed to satisfy the minimum gain constraint for all values in the uncertainty ellipsoid. A. Lagrange multiplier methods It is natural to suspect that we may compute the RMVB efficiently using Lagrange multiplier meth ods. See, for example, 10] 17] 18] 19, 12.1. 1] and [20]. Indeed this is the case. The RMVB is the optimal solution of minimize xTRx subject to IIATII = cTx 17 (19) if we impose the additional constraint that cTx 1. We define the Lagrangian L: R n x R R associated with (19) as: L( R (11A I 17) xT(R Q)x 2cx , 20) where Q ....

D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Athena Scientific, Belmont, Mass., 1996.

....= 8.336 N, 1.32733102m3 s 2 (gravitational constant for the Sun) and tf = 193 days. The trajectory, appearing in Figure 1, takes the spacecraft from an Earth orbit around the Sun to a Mars orbit. The terminal constraints on u and v at tf were treated using penalty multiplier techniques (see [2] and [34] We discretized the problem using the 3 stage methods (a) and (b) To estimate the errors associated with each discretization, solutions were obtained for three meshes corresponding to N = 500, 1000, and 2000, and Aitken s extrapolation was used to estimate the exact solution at each ....

D. P. BERTSEKAS, Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, 1982.

....solvers is far from trivial, due to the intrinsic non differentiable nonlinearity of the problem. Regularization techniques [7, 11, 13, 14] require careful handling of regularization parameters in order to find a reasonable compromise between efficiency and accuracy. Dual techniques (cf. e.g. [5, 13, 14, 15]) are based on saddle point formulations incorporating the constraints by means of Lagrange multipliers. Active set strategies [2, 16, 18, 20] iteratively provide approximations of the contact set. A linear subproblem with given contact set has to be solved in each iteration step and multigrid ....

D.P. Bertsekas. Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York, 1982.

.... = N k k h g , 0 1 = N k I I k h k g (3) The constraints (3) give the prediction of the polynomial degrees 0, I, and from them can closed form solutions for the FIR coefficients for low degree polynomial input signals be solved by the method of Lagrange multipliers [9]. The closed form solutions for FIR coefficients for the 1st, 2nd, and 3rd degree polynomial input signals can be found in [8] 2.2. Polynomial Predictive FIR Differentiators PPFDs are derived in the similar way as the PFPs. For the PPFDs, the filter input output relation is written as [5,7] ....

D. Bertsekas, Constrained Optimization and Lagrange Multipliers Methods. New York, NY, USA: Academic Press, 1982, Chapter 1.

....of the form (1.1) the multipliers method recently proposed by Vicente [9] for nonlinear optimization problems of the form min f(y; u) s.t. c(y; u) 0; y; u) 0: 1.2) Here it is assumed that the partial Jacobian of c with respect to y is square and invertible. The traditional multipliers method [1] is based on the augmented Lagrangian penalty function and involves an update formula of the form k 1 = k 1 k h(x k ) Note that the Lagrangian term of the augmented Lagrangian penalty function involves only the equality constraints h(x) 0. The multipliers corresponding to h(x) ....

....of the new multipliers method for general programming problems of the form (1.1) The techniques we use here are similar to what has been used in [9] for the problem class (1. 2) which in turn shares a lot in common with the proof of local convergence of the traditional multipliers method [1]. It is shown that the neighborhood of local convergence is smaller than in the traditional multipliers method, see (4.5) On the other hand we analyze for the rst time the global convergence behavior of the new multipliers method. The analysis is carried out for (1.1) but it is easily applied to ....

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D. P. Bertsekas, Constrained optimization and Lagrange multiplier methods, Computer Science and Applied Mathematics, Academic Press, New York, 1982.

....open set in R n , F : V # R m be a given mapping. Consider an equalityconstrained optimization problem minimize f(x) subject to x # D, 41) where the (feasible) set D is defined in (1) and f : V # R is a given (objective) function. Similar to the standard exterior penalty framework [12,7,28], define the family of functions # #, # : V # R,# #, # (x) f(x) ##F(x)# # , 42) where # 0 is the penalty parameter, and # 0 is the power of the penalty function. For simplicity, we shall assume # to be fixed. Consider the family of (unconstrained) optimization problems minimize # ....

Bertsekas, D.P. (1982): Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York

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Bertsekas, D. P., Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, NY, 1982.

....and (c) of Assumption A are constraint qualification conditions, which are needed to guarantee the existence of a Kuhn Tucker vector for the problem (see [Roc70] p. 277) We now describe the exponential multiplier method proposed by Kort and Bertsekas [KoB72] for solving the problem (see also [Ber82], Sec. 5.1.2) Let # : # # # be the exponential penalty function given by # (t) e t 1. 1.2) We associate a multiplier j 0 with the jth constraint. The method performs a sequence of unconstrained minimizations, and iterates on the multipliers at the end of each minimization. At the kth ....

....to what happens in usual exterior penalty methods [FiM68] Lue84] and for this reason, much of the standard analysis for exterior penalty and multiplier methods cannot be applied to the exponential method of multipliers. It can be shown that the minimum in Eq. 1. 3) is attained for all k ([Ber82], p. 337) For a brief justification, note that if this minimum were not attained, then f and the functions g j would share a direction of recession, in which case the optimal solution set of (P) is unbounded (see [Roc70] Section 8) thus contradicting Assumption A. We will consider two rules for ....

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Bertsekas, D. P., Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, N. Y., 1982.

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Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York (1982)

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Bertsekas, D. P., "Constrained Optimization and Lagrange Multiplier Methods", Academic Press, 1982

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D.P. Bertsekas. Constrained optimization and Lagrange multiplier methods. Academic Press, 1982.

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D.P. Bertsekas. Constrained Optimization and Lagrange Multiplier Methods. Academic Press, 1982.

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D. P. Bertsekas. Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York, 1982.

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D. P. Bertsekas [1982], Constrained optimization and Lagrange multiplier methods, Academic Press, New York.

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D P Bertsekas. Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York, 1982.

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D.P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Academic Press, 1982. (a) (b) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

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D. Bertsekas. Constrained Optimization and Lagrange Multiplier Methods. Athena Scientific, Belmont, Mass., 1996.

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Dimitri P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Computer Science and Applied Mathematics, Academic Press, 1982.

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D. P. Bertsekas, Constrained optimization and Lagrange multiplier methods, Academic Press, New York, 1982.

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D. P. Bertsekas, Constrained Optimization and Lagrange Multipliers Methods, Academic Press, London, 1982.

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D.P. Bertsekas, 1982. Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York.

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D. P. Bertsekas, Constrained optimization and Lagrange multiplier methods, ser. Computer Science and Applied Mathematics. New York: Academic Press Inc., 1982.

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D. P. Bertsekas. Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York, 1982.

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D.P. Bertsekas. Constrained Optimization and Lagrange Multiplier Methods. Academic Press (New York 1982).

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D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Computer Science and Applied Mathematics, Academic Press, New York, 1982.

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Bertsekas, D.P. (1982). Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York.

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D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York,

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Bertsekas, D. P., Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, New York, 1982.

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D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods. Reading, MA: Academic, 1982.

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D. Bertsekas. Constrained Optimization and Lagrange Multiplier Methods. Athena Scientific, 1996.

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D. Bertsekas, Constrained optimization and Lagrange multiplier methods, Academic Press, New York (1982).

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D. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York (1982).

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D. Bertsekas, Constrained optimization and Lagrange multiplier methods, Academic Press, New York (1982).

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D. Bertsekas, Constrained Optimization and Lagrange Multipliers Methods. New York, NY, USA: Academic Press, 1982.

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