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A Sensitivity Analysis and a Convergence Result for a Sequential.. - Freund (2003) (1 citation) (Correct)
.... v) r x (B(x) ffl Y ) u i r x c i (x) v j r x d j (x) 15) Note that in (14) the gradients of the vector valued functions c(x) and d(x) are defined as r x c(x) D x c(x) and r x d(x) D x d(x) If the problem (12) is convex and satisfies Slater s condition [13], then for each optimal solution x of (12) there exists an m Theta m matrix Y 0 and vectors u 2 IR , u 0, and v 2 IR such that the quadruple (x; Y; u; v) is a saddle point of the Lagrangian (13) L. More generally, for nonconvex problems (12) let x 2 IR be a feasible point of (12) ....
....Y 0 and vectors u 2 IR , u 0, and v 2 IR such that the quadruple (x; Y; u; v) is a saddle point of the Lagrangian (13) L. More generally, for nonconvex problems (12) let x 2 IR be a feasible point of (12) and assume that the Robinson or Mangasarian Fromovitz constraint qualification [13, 18, 19] is satisfied at x, i.e. the matrix D x d(x) has full rank and there exists a vector Deltax 6= 0 such that B(x) D x B(x) Deltax] OE 0, c(x) D x c(x) Deltax 0, and D x d(x) Deltax = 0. Then, if x is a local minimizer of (12) the first order optimality condition is satisfied, i.e. there ....
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Mangasarian, O.L. (1969): Nonlinear Programming. MvGraw-Hill, New York
Parallel Variable Distribution for Constrained Optimization - Sagastizabal, Solodov (Correct)
....developing a stopping rule for solving (3.14) or, equivalently, an approximation criterion for projection directions to be used in the PVD scheme. For algorithmic purposes, it is important to make this criterion constructive and implementable. Assuming some constraint quali cation condition [13], we have that z solves (3.14) i.e. z = P C [x rf(x) if, and only if, the pair ( z; u) 2 satis es the KKT system r z L( z; u) z x rf(x) c u = 0 ; c( z) 0 ; u 0 and h u; c( z)i = 0 ; 3.15) L(z; u) 1 hu; c(z)i (3.16) is the standard Lagrangian ....
Mangasarian, O.: 1969, Nonlinear Programming. New York: McGraw{Hill.
Gilding the Lily: a Variant of the Nelder-Mead Algorithm.. - Nazareth, Tseng (2001) (Correct)
....such a triple with either x 2 or x e , depending on whether f 1 f oc . The analysis of Algorithm 2 exploits the fact that this algorithm may be viewed as a special case of the method in [16] so that the convergence result therein may be applied. Recall that f is said to be quasiconvex on [8] if f(x (y x) maxff(x) f(y)g 8 2 [0; 1] 8x; y 2 : Notice that quasiconvexity generalizes unimodality to the multivariable case and it is weaker than convexity. Theorem 2 Assume that f is continuously di erentiable and quasiconvex on and is uniformly continuous on fx 2 : ....
O. L. Mangasarian, Nonlinear Programming, McGraw-Hill, New York, NY, 1969.
Optimality Conditions For Irregular - Inequality-Constrained Problems.. (Correct)
....and an example. In this section, we provide a comparison of the results obtained above with known approaches to irregular problems, and illustrate our development by an example. First, we mention Abadie s and Kuhn Tucker s constraint qualifications (CQs) for nonlinear programming (see [22]; there are also some other CQs of similar type) These are weaker than the Mangasarian Fromovitz constraint qualification (MFCQ) but still guarantee that the tangent cone is given by the linearized model of the constraints; e.g. see [23, 22] From the point of view of the problem data, these ....
.... qualifications (CQs) for nonlinear programming (see [22] there are also some other CQs of similar type) These are weaker than the Mangasarian Fromovitz constraint qualification (MFCQ) but still guarantee that the tangent cone is given by the linearized model of the constraints; e.g. see [23, 22]. From the point of view of the problem data, these CQs are less constructive than MFCQ, which is closer to our development. MFCQ is subsumed by our framework. Such CQs of nonalgebraic nature are usually rather di#cult to verify directly. Perhaps even more importantly, we deal here with a more ....
O. L. Mangasarian, Nonlinear Programming, McGraw--Hill, New York, 1969.
Capacity Scaling and Spectral Efficiency in Wideband Correlated.. - Vasanthan (2002) (1 citation) (Correct)
.... by i n 4 (117) Setting partial derivatives to zero we get 4 n 4 (118) Observe ] is a solution to (118) It is also easy to check that this solution together with associated satisfies the Kuhn Tucker condition [31], and thus achieves the maximum which turns out to be . 38 APPENDIX II PROOF OF LEMMA 4 h 11 0 h 22 h 32 0 h 23 h 33 0 h 34 y 1 y 2 y 3 y 4 Fig. 6. A schematic illustrating Rayleigh sub channel construction from a dimensional connected channel. s and s are input ....
O. L. Mangasarian, Nonlinear Programming, McGraw-Hill, New York, 1994.
Multiplier Rules under Mixed Assumptions of Differentiability and.. - Ye (Correct)
.... requires that the objective function and the inequality constraints be differentiable, the equality constraints be continuously differentiable at the optimal solution, and the abstract constraint set be convex with nonempty interior (e.g. see Bazaraa, Sherali, and Shetty [1] and Mangasarian [14]) Over the last three decades, the classical multiplier rule was extended under two different assumptions: differentiability and Lipschitz continuity. On the one hand, the classical multiplier rule was extended in the direction of eliminating the smoothness assumption while keeping the ....
O.L. Mangasarian, Nonlinear Programming, McGraw-Hill, New York,
Solving Nonconvex Problems of Multibody Dynamics with Contact .. - Anitescu, Hart (Correct)
....contacts while keeping feasibility of the joint constraints. This condition can be estimated visually for most simple configurations. When a nonlinear program whose constraints are ( nd (2) sis ties he above relation, i is said o satisfy he Mangasarian Fromoviz consrain qualification or MFCQ [14, 18]. This propery is essential to ensure the good behavior (Lipschiz continuity) of the solution of the nonlinear program with respect o its parameters [19] Lipschiz continuity is the key element that allows us to show convergence of our fixed point iteration (successive convex relaxation) S ....
Mangasarian, O. L., Nonlinear Programming, McGraw-Hill, New York 1969.
Duality, Geometry, and Support Vector Regression - Bi, Bennett (2001) (Correct)
....# tube) A hard # tube exists for a given # 0 if and only if the following system in (u, v) has no solution: X # u = X # v, e # u = e # v = 1, y #e) # u #e) # v 0, u 0. 3) Proof A hard # tube exists if and only if System (2) has a solution. By Gale s Theorem of the alternative [4], system (2) has a solution if and only if the following alternative system has no solution: X # u = X # v, e # u = e # v, y #e) # u (y #e) # v = 0. Rescaling by # where # = e # u = e # v 0 yields the result. We use the following definitions of separation of convex sets. ....
....2: The solution # tube found by C SVR can have # #. Squares are original data. Dots are in . Triangles are in . Support Vectors are circled. w,#,#,# 1 #) s.t. Xw #(y #e) Xw #(y 0. 5) Dual C SVR (4) can be derived by taking the Wolfe or Lagrangian dual [4] of primal C SVR (5) and simplifying. We prove that the optimal plane from C SVR bisects the # tube. The supporting planes for class and class determines the lower and upper edges of the # tube respectively. The support vectors from correspond to the points along the lower and ....
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O. Mangasarian. Nonlinear Programming. SIAM, Philadelphia, 1994.
Characterizing Uncertainty Aversion through Preference for.. - Klibanoff (1996) (1 citation) (Correct)
....of probability vectors satisfying a set of 7 linear inequalities. This is done in the lemma below. The only task remaining is then to characterize existence. To do this I apply a well known result from the theory of linear inequalities, Motzkin s Theorem of the alternative (see e.g. Mangasarian [16]) The existence of a solution to the resulting alternative system is then (after a bit of rearrangement) shown to be equivalent to the conditions of the theorem. Let the vector of utility payos to the act f be denoted F(fu(f(s) g) and similarly for G. The following lemma reduces the conditions ....
....(1) with the condition hi p=0 e e and the condition p 0 with the equivalent Ip 0 where I is a 2n x 2n identity matrix. To summarize, we would like to characterize when there exists a p such that (a)Ap 0 (b)Ip 0 and hi (c)p=0: e e By Motzkin s Theorem of the alternative (Mangasarian [16]) either (a) b) and (c) has a solution p or hi0 00 Ay Iy y =0 e e 13 4 ( y 0;y 0 13 has a solution y , y , y , but never both. Note that y 0 means that each element of y 134 1 1 is greater than or equal to zero with at least one element strictly positive. y 0 means 3 almost the ....
Mangasarian, O. (1969) Nonlinear Programming. McGraw-Hill.
Duality, Geometry, and Support Vector Regression - Jinbo Bi And (Correct)
....A hard # tube exists for a given # 0 if and only if the following system has no solution: X # u = X # v, e # u = e # v = 1, y #e) # u (y #e) # v 0, u # 0, v # 0. 3) Proof A hard # tube exists if and only if System (2) has a solution. By Gale s Theorem of the alternative [4], system (2) has a solution if and only if the following alternative system has no solution: X # u = X # v, e # u = e # v, y #e) # u (y #e) # v = 1, u # 0, v # 0. Rescaling by 1 # where # = e # u = e # v 0 yields the result. Note that if # # # 0 then (y #e) # u (y ....
....yields the regression function. As in classification [1] the convex hull form is the dual problem. Primal C SVR is: min w,#,#,# 1 2 #w# 2 1 2 # 2 (# #) s.t. Xw #(y #e) #e # 0 Xw #(y #e) #e # 0. 5) Dual C SVR (4) can be derived by taking the Wolfe dual [4] of primal C SVR (5) and converting the dual maximization problem to a minimization problem. We prove that the optimal plane from C SVR bisects the # tube. The supporting planes for class D and class D determines the lower and upper edges of the # tube respectively. The support vectors ....
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O. Mangasarian. Nonlinear Programming. SIAM. 1994.
Multicategory Classification by Support Vector Machines - Bredensteiner, Bennett (1999) (2 citations) (Correct)
....one nonzero variable u ij l , j = 1, k, j #= i. In Figure 8, support vectors are represented by a circle around the point. Some points have double circles which indicate that two dual variables u ij l 0, j = 1, 3, j #= i. By the complementarity within the KKT conditions [14], u ij l 0 # A i l (w i w j ) # i # j ) 1. Consequently the support vectors are located closest to the separating function. In fact, the remainder of the points, those that are not support vectors, are not necessary in the construction of the separating function. The ....
O. L. Mangasarian. Nonlinear Programming. McGraw--Hill, New York, 1969.
Games Played through Agents - Prat, Rustichini (2001) (1 citation) (Correct)
....8 Theorem 3 (Farkas) Exactly one of the following alternatives is true: a) There exists asolutionx to the linear system of (in)equalities given by Ax # a and Bx = b;or(b) There exist vectors and # such that: i) A #B =0; ii) # 0; and (iii) a #b 0. 8 See for instance Mangasarian [#4]. #4 We now apply Farkas Lemma to (9) and (#0) Recall that d is a vector of MN elements. We construct a matrix A of dimensions MSMN,amatrixB of dimensions N MN,avectora with MS elements, and a vector b with N elements. For m, j # M , i, n # N , s # S,let A (ms,jn) # if j 6= ....
Olvi L. Mangasarian. Nonlinear Programming. McGraw-Hill, #969.
On the Global Convergence of a Filter-SQP Algorithm - Fletcher, Leyffer, Toint (2000) (2 citations) (Correct)
....loss of generality we may assume that they also satisfy bounds krf(x)k 2 M , krc i (x)k 2 M , i 2 E [ I, for all x 2 X. Our global convergence theorem concerns Kuhn Tucker (KT) necessary conditions under a Mangasarian Fromowitz constraint quali cation (MFCQ) see for example, Mangasarian [8]) This is essentially an extended form of the Fritz John conditions for a problem that includes equality constraints. A feasible point x of problem P satis es MFCQ if and only if both (i) the vectors a i , i 2 E are linearly independent, and (ii) there exists a direction s that satis es s ....
Mangasarian, O.L. (1969), Nonlinear Programming, McGraw-Hill, New York.
Error bounds for 2-regular mappings with Lipschitzian.. - Izmailov, Solodov (2001) (Correct)
....no q # R m such that the following system has a solution in h # R m : #q, h# 0, # g # i (x) h # = 0, h i # 0, i # A, # g # i (x) h # # 0, h i = 0, i # B, # g # i (x) h # = 0, i # I 1 , h i = 0, i # I 2 . By the Motzkin theorem of the alternatives [26], the latter is equivalent to the following system (in #, having a solution for all q # R m : # i#A # i e i # i#B # i g # i (x) # i#A#I 1 i g # i (x) # i#B#I 2 i e i = q, # i # 0 , i # A # B = I 0 . In other words, R m = cone e i i # A cone ....
Mangasarian, O.L. (1969): Nonlinear Programming. McGraw--Hill, New York
Efficient orthogonal factorization for constraint.. - Alvarado, Orfanogianni (1998) (Correct)
....Spot Pricing are discussed. Keywords: QR Decomposition, Complex QR, Factor update,Quadratic Programming 1 Introduction This paper addresses the problem of solving optimization problems (particularly quadratic programming, or QP problems) Quadratic programming is a well developed science [1]. Consider first a QP problem subject exclusively to linear equality constraints: Minimize 1 2 x # Hx c # x subject to Ax = B (1) where H a symmetric matrix, assumed to be positive definite (convex QP) The solution to this problem by the augmented method can be obtained from: # HA # A0 ## ....
O. Mangasarian. Nonlinear Programming. SIAM, 1994.
Mathematical Programming Approaches To Machine Learning And Data.. - Bradley (1998) (1 citation) (Correct)
.... determined by (26) generate a strictly decreasing sequence of objective function values for the FSV 32 problem (13) and terminate at an iteration i with a stationary point (which may also be a global minimum solution) that satisfies the following minimum principle necessary optimality criterion [98]. 1 Gamma ) e 0 m (y Gamma y i ) e 0 k (z Gamma z i ) ff( Gammaffv i ) 0 (v Gamma v i ) 0; 8 feasible (w; fl; y; z; v) 28) We solve the FSV problem (13) by Algorithm 2.1.3. We now turn our attention to solving the FSB problem (16) Bilinear Algorithm Due to ....
....also easily be handled by the formulations (7) 12) 13) 16) 34) and (35) 96] If the data are mapped nonlinearly via Phi : R n R , a nonlinear separating surface in R n is easily computed as a linear separator in R . In practice, one usually solves (36) by way of its dual [98]. In this formulation, the data enter only as inner products which are computed in the transformed space via a kernel function K(x; y) Phi(x) Delta Phi(y) 33, 161, 164] We note that separation errors in (34) 36) are weighted equally conforming to the SVM formulations in [33, 161] In ....
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O. L. Mangasarian. Nonlinear Programming. McGraw--Hill, New York,
Clustering via Concave Minimization - Bradley And Mangasarian (1997) (7 citations) Self-citation (Mangasarian) (Correct)
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O. L. Mangasarian. Nonlinear Programming. SIAM, Philadelphia, PA, 1994.
A Newton Method for Linear Programming - Mangasarian (2002) Self-citation (Mangasarian) (Correct)
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O. L. Mangasarian. Nonlinear Programming. SIAM, Philadelphia, PA, 1994. 18
A Feature Selection Newton Method for Support Vector Machine .. - Fung, Mangasarian (2002) (2 citations) Self-citation (Mangasarian) (Correct)
....approach zero for standard penalty application for solving the dual linear program (2.8) However, in our approach we shall establish the fact that e will remain finite and we still can obtain an exact solution to our linear programming SVM (2. 6) To do that we first write the Karush Kuhn Tucker [13] necessary and sufficient optimality conditions for the penalty problem (2.9) 2.15) O u I ( e DA(A Du e) w p q (2.10) OA( A O e) DeeDu (u pc) O, y p (A Du e) q ( A Du e) 7 eDu, and make use of the simple equivalence: a b z Oa (a b)O, then (2.13) together with the ....
O. L. Mangasarian. Nonlinear Programming. SIAM, Philadelphia, PA, 1994.
Data Selection for Support Vector Machine Classifiers - Fung, Mangasarian (2000) (5 citations) Self-citation (Mangasarian) (Correct)
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O. L. Mangasarian. Nonlinear Programming. SIAM, Philadelphia, PA, 1994.
Active Support Vector Machine Classification - Mangasarian, Musicant (2000) (1 citation) Self-citation (Mangasarian) (Correct)
....error variable S: Aiw Yi 7 1, for Dii 1, Aiw Yi 1, for Dii 1. 5) Traditionally the 1 norm of the error variable y is minimized parametrically with weight v in (2) resulting in an approximate separation as depicted in Figure 1. The dual to the standard quadratic linear SVM (2) [13, 22, 14, 7] is the following: rain lu DAA Du e u s.t. e Du 0, 0 u re. 6) uC R 2 (4) can be obtained from the solution of the dual problem above [15, Eqns. 5 and 7] We note immediately that the matrix DAA D appearing in the dual objective function (6) is not positive definite in general because ....
....that is with respect to both orientation and location of the planes, rather that just with respect to w which merely determines the orientation of the plane. This leads to the following reformulation of the SVM: min vY (ww ) s.t.D(Aw e ) y e. 7) v,7,y)CRn the dual of this problem is [13]: rain I .I 0 cR.u ( D(AA cc )D)u c . 8) The variables (w, of the primal problem which determine the separating surface (4) are recovered directly from the solution of the dual (8) above by the relations: w n D, v 7 (9) We immediately note that the matrix appearing in the dual ....
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O. L. Mangasarian. Nonlinear Programming. SIAM, Philadelphia, PA, 1994.
Proximal Support Vector Machine Classifiers - Fung, Mangasarian (10 citations) Self-citation (Mangasarian) (Correct)
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O. L. Mangasarian. Nonlinear Programming. SIAM, Philadelphia, PA, 1994.
Finite Newton Method for Lagrangian Support Vector Machine.. - Fung, Mangasarian (2002) Self-citation (Mangasarian) (Correct)
....our modified SVM problem as follows: 4 rain y y (w w 2) y)cw TM s.C. D(Aw e) y e (6) y o. It has been shown computationally [17] that this reformulation (6) of the conventional support vector machine formulation (2) yields similar results to (2) The dual of this problem is [12]: OuR TM 2 The variables (w, if) of the primal problem which determine the separating surface (4) are recovered directly from the solution of the dual (7) above by the relations: w ADu, y , eDu. 8) We immediately note that the matrix appearing in the dual objective function is positive ....
O. L. Mangasarian. Nonlinear Programming. SIAM, Philadelphia, PA, 1994.
Knowledge-Based Support Vector Machine Classifiers - Fung, Mangasarian, Shavlik (2002) (2 citations) Self-citation (Mangasarian) (Correct)
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O. L. Mangasarian. Nonlinear Programming. SIAM, Philadelphia, PA, 1994.
Set Containment Characterization - Mangasarian Self-citation (Mangasarian) (Correct)
....i ) Bx # b # 0. 3) Proof ( i) # (iii) For i # 1, m , the m linear programs of (3) are feasible because B #= # and their objective functions are bounded below by zero and hence attain their nonnegative minima as asserted by (3) iii) # (ii) By linear programming duality [3, 11], for i = 1, m, each of the m linear programs that are dual to the m linear programs of (3) are solvable and satisfy: max u ( b # u a i ) B # u = A # i , u # 0 # 0. 4) Calling the solution of each of these m dual linear programs u i # R k , i = 1, m, and ....
O. L. Mangasarian. Nonlinear Programming. SIAM, Philadelphia, PA, 1994.
Loss Functions and Structured Domains for Support Vector Machines - Portera (2005) (Correct)
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Olvi L. Mangasarian. Nonlinear Programming. SIAM, Philadelphia, 1994.
Journal of Machine Learning Research 7 (2006) 603--624.. - Algorithms Mingrui Wu (Correct)
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O. L. Mangasarian. Nonlinear Programming. McGraw-Hill, New York, 1969.
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O. L. Mangasarian, Nonlinear programming, McGraw--Hill, New York, 1969.
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O. Mangasarian. Nonlinear Programming. McGraw-Hill, New York, 1969.
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Olvi L. Mangasarian. Nonlinear Programming, McGraw Hill, New York (1969). 102
Soft Margins for AdaBoost - Rätsch, Onoda, Müller (1998) (Correct)
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O.L. Mangasarian. Nonlinear Programming. McGraw-Hill, New York, NY, 1969.
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Minimum Principle of Complementary Energy in Nonlinear.. - Kanno, Ohsaki (2002) (Correct)
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O.L. Mangasarian, Nonlinear Programming, McGraw-Hill, New York, 1969.
An Axiomatic Model of Non-Bayesian Updating - Epstein (2004) (Correct)
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O.L. Mangasarian, Nonlinear Programming, McGraw-Hill, New York: 1969.
The Kernel Mutual Information - Gretton, Herbrich, Smola (2003) (Correct)
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O. Mangasarian. Nonlinear Programming. SIAM, Philadelphia, 1994.
AMO - Advanced Modeling and Optimization, Volume 5, Number 3.. - Class Of Methods (Correct)
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O.L.Mangasarian, (1984) Nonlinear Programming, SIAM. 220
Soft Margins for AdaBoost - Gunnar Ratsch Gmd (1998) (Correct)
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O.L. Mangasarian. Nonlinear Programming. McGraw-Hill, New York, NY, 1969.
Non-Bayesian Updating: A Theoretical Framework - Epstein, Sandroni (2003) (Correct)
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O.L. Mangasarian, Nonlinear Programming, McGraw-Hill, New York: 1969.
Efficient Computation of Equilibrium Prices for Markets.. - Codenotti, Varadarajan (Correct)
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O. L. Mangasarian. Nonlinear Programming, McGraw-Hill, 1969.
A Fixed-Point Iteration Approach for Multibody Dynamics With.. - Anitescu, Hart (2004) (Correct)
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Mangasarian, O.L.: Nonlinear Programming. McGraw-Hill, NewYork, 1969
Solving Nonconvex Problems of Multibody Dynamics with Joints, .. - Anitescu, Hart (Correct)
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Mangasarian, O. L., Nonlinear Programming, McGraw-Hill, New York 1969.
A Constraint-Stabilized Time-Stepping Approach for Rigid.. - Anitescu, Hart (2002) (Correct)
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Mangasarian, O. L., Nonlinear Programming, McGraw-Hill, New York 1969.
A General Framework for Convex Relaxation of Polynomial.. - Kojima, Kim, Waki (2002) (2 citations) (Correct)
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O. L. Mangasarian, Nonlinear Programming, McGraw-Hill, New York, 1969.
On Approximations With Finite Precision In Bundle Methods For.. - Solodov (2002) (Correct)
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O.L. Mangasarian. Nonlinear Programming. McGraw{Hill, New York, 1969.
Decomposition Algorithms for Stochastic Programming on a.. - Linderoth, Wright (2001) (3 citations) (Correct)
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O. L. Mangasarian. Nonlinear Programming. McGraw-Hill, New York, 1969.
The Dual Active Set Algorithm and the Iterative Solution of Linear .. - Hager (2003) (Correct)
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O. L. Mangasarian, Nonlinear Programming, McGraw-Hill, New York, 1969.
An Axiomatic Model of Non-Bayesian Updating - Epstein (2003) (Correct)
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O.L. Mangasarian, Nonlinear Programming, McGraw-Hill, New York: 1969.
Decomposition Algorithms for Stochastic Programming on a.. - Linderoth, Wright (2002) (3 citations) (Correct)
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O. L. Mangasarian. Nonlinear Programming. McGraw-Hill, New York, 1969.
Alternating Directions Methods for the Parallel.. - Spyridon.. (Correct)
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O.L. Mangasarian. Nonlinear programming. McGraw-Hill, 1969.
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