Optimizing Convex Functions Over Non-Convex Domains - Columbia University
Optimizing Convex Functions over Non-Convex Domains Daniel Bienstock and Alexander Michalka Columbia University Berlin 2012 Bienstock, Michalka Convex obj non-convex domain Columbia
Introduction Generic Problem: min Q(x), Bienstock, Michalka Convex obj non-convex domain s.t. x F, Columbia
Introduction Generic Problem: min Q(x), s.t. x F, Q(x) convex, especially: convex quadratic Bienstock, Michalka Convex obj non-convex domain Columbia
Introduction Generic Problem: min Q(x), s.t. x F, Q(x) convex, especially: convex quadratic F nonconvex Bienstock, Michalka Convex obj non-convex domain Columbia
Introduction Generic Problem: min Q(x), s.t. x F, Q(x) convex, especially: convex quadratic F nonconvex Examples: F is a mixed-integer set Bienstock, Michalka Convex obj non-convex domain Columbia
Introduction Generic Problem: min Q(x), s.t. x F, Q(x) convex, especially: convex quadratic F nonconvex Examples: F is a mixed-integer set F is constrained in a nasty way, e.g. x1 3 sin(x2 ) 2 cos(x3 ) 4 Bienstock, Michalka Convex obj non-convex domain Columbia
Or, Convex constraint: Q(x) q, and x F , Q(x) convex, especially: convex quadratic F nonconvex Examples: F is a mixed-integer set F is constrained in a nasty way, e.g. x1 3 sin(x2 ) 2 cos(x3 ) 4 Bienstock, Michalka Convex obj non-convex domain Columbia
Exclude-and-cut min z, s.t. z Q(x), x F 0. Let F̂ be a convex relaxation of F Bienstock, Michalka Convex obj non-convex domain Columbia
Exclude-and-cut min z, s.t. z Q(x), x F 0. Let F̂ be a convex relaxation of F 1. Let (x , z ) argmin{ z : z Q(x), x F̂ } Bienstock, Michalka Convex obj non-convex domain Columbia
Exclude-and-cut min z, s.t. z Q(x), x F 0. Let F̂ be a convex relaxation of F 1. Let (x , z ) argmin{ z : z Q(x), x F̂ } 2. Find an open set S s.t. x S and S F . Examples: lattice-free sets, geometry Bienstock, Michalka Convex obj non-convex domain Columbia
Exclude-and-cut min z, s.t. z Q(x), x F 0. Let F̂ be a convex relaxation of F 1. Let (x , z ) argmin{ z : z Q(x), x F̂ } 2. Find an open set S s.t. x S and S F . Examples: lattice-free sets, geometry 3. Add to the formulation an inequality az αT x α0 valid for { (x, z) : x Rn S, z Q(x) } but violated by (x , z ). Bienstock, Michalka Convex obj non-convex domain Columbia
The SUV problem given full-dimensional polyhedra P 1 , . . . , P K in Rd , find a point closest to the origin not contained inside any of the P h . Bienstock, Michalka Convex obj non-convex domain Columbia
The SUV problem given full-dimensional polyhedra P 1 , . . . , P K in Rd , find a point closest to the origin not contained inside any of the P h . min kxk2 s.t. x Rd K [ int(P h ), h 1 Bienstock, Michalka Convex obj non-convex domain Columbia
The SUV problem given full-dimensional polyhedra P 1 , . . . , P K in Rd , find a point closest to the origin not contained inside any of the P h . min kxk2 s.t. x Rd K [ int(P h ), h 1 (application: X-ray lythography; see Ahmadia (2010)) (yes, it is NP-hard) Bienstock, Michalka Convex obj non-convex domain Columbia
Bienstock, Michalka Convex obj non-convex domain Columbia
Typical values for d (dimension): less than 20; usually even smaller Typical values for K (number of polyhedra): possibly hundreds, but often less than 50 Very hard problem Bienstock, Michalka Convex obj non-convex domain Columbia
First problem setting Let Q(x) be a strongly convex function on Rd , Let P Rd be such that each connected component is homeomorphic to a (positive radius) ball or a half-plane so, closed and nonempty interior We also assume that Rd P 6 . Bienstock, Michalka Convex obj non-convex domain Columbia
First problem setting Let Q(x) be a strongly convex function on Rd , Let P Rd be such that each connected component is homeomorphic to a (positive radius) ball or a half-plane so, closed and nonempty interior We also assume that Rd P 6 . Want to produce a linear inequality description for: n o (x, q) Rd 1 : Q(x) q, x Rd int(P) . Bienstock, Michalka Convex obj non-convex domain Columbia
First-order cut: Given y Rd , q Q(y ) T Q(y )(x y ) Bienstock, Michalka Convex obj non-convex domain Columbia
First-order cut: Given y Rd , q Q(y ) T Q(y )(x y ) is valid for all x Rd , Bienstock, Michalka Convex obj non-convex domain Columbia
First-order cut: Given y Rd , q Q(y ) T Q(y )(x y ) is valid for all x Rd , does not cut anything off Bienstock, Michalka Convex obj non-convex domain Columbia
First-order cut: Given y Rd , q Q(y ) T Q(y )(x y ) is valid for all x Rd , does not cut anything off How can we use the structure of P to strengthen the inequality? Bienstock, Michalka Convex obj non-convex domain Columbia
Definition: Given y P, say P is locally flat at y if B(µ, ρ) P with kµ y k2 ρ and ρ 0. ρ µ P y d R P Bienstock, Michalka Convex obj non-convex domain Columbia
Suppose P is locally flat at y . Write: a µ y . Bienstock, Michalka Convex obj non-convex domain Columbia
Suppose P is locally flat at y . Write: a µ y . Lemma: For real α 0 small enough the inequality q Q(y ) T Q(y )(x y ) αaT (x y ) is valid, Bienstock, Michalka Convex obj non-convex domain Columbia
Suppose P is locally flat at y . Write: a µ y . Lemma: For real α 0 small enough the inequality q Q(y ) T Q(y )(x y ) αaT (x y ) is valid, i.e., for all (x, q) with x Rd P and q Q(x). Bienstock, Michalka Convex obj non-convex domain Columbia
Suppose P is locally flat at y . Write: a µ y . Lemma: For real α 0 small enough the inequality q Q(y ) T Q(y )(x y ) αaT (x y ) is valid, i.e., for all (x, q) with x Rd P and q Q(x). Inequality is tight at (y , Q(y )), and cuts-off points (x, Q(x)) and x int(P). Bienstock, Michalka Convex obj non-convex domain Columbia
Suppose P is locally flat at y . Write: a µ y . Lemma: For real α 0 small enough the inequality q Q(y ) T Q(y )(x y ) αaT (x y ) is valid, i.e., for all (x, q) with x Rd P and q Q(x). Inequality is tight at (y , Q(y )), and cuts-off points (x, Q(x)) and x int(P). Largest possible α: “lifted first-order inequality”. Bienstock, Michalka Convex obj non-convex domain Columbia
n o (x, q) Rd 1 : Q(x) q, x Rd int(P) Bienstock, Michalka Convex obj non-convex domain Columbia
n o (x, q) Rd 1 : Q(x) q, x Rd int(P) Theorem. Any linear inequality valid for S is dominated by a lifted first-order inequality. More precisely, conv (x, q) Rd 1 : Q(x) q, x Rd int(P) {(x, q) Rd 1 : LFO FO } Bienstock, Michalka Convex obj non-convex domain Columbia
n o (x, q) Rd 1 : Q(x) q, x Rd int(P) Theorem. Any linear inequality valid for S is dominated by a lifted first-order inequality. More precisely, conv (x, q) Rd 1 : Q(x) q, x Rd int(P) {(x, q) Rd 1 : LFO FO } How do we make this computationally practicable? Bienstock, Michalka Convex obj non-convex domain Columbia
First problem setting Let Q(x) is a positive definite quadratic on Rd , P {x Rd : aiT x bi , 1 i m} full dimensional Bienstock, Michalka Convex obj non-convex domain Columbia
First problem setting Let Q(x) is a positive definite quadratic on Rd , P {x Rd : aiT x bi , 1 i m} full dimensional Want to produce a linear inequality description for: n o (x, q) Rd 1 : Q(x) q, x Rd int(P) . Bienstock, Michalka Convex obj non-convex domain Columbia
First problem setting Let Q(x) is a positive definite quadratic on Rd , P {x Rd : aiT x bi , 1 i m} full dimensional Want to produce a linear inequality description for: n o (x, q) Rd 1 : Q(x) q, x Rd int(P) . change in coordinates d X . S (x, q) Rd 1 : xj2 q, x Rd int(P) . j 1 Bienstock, Michalka Convex obj non-convex domain Columbia
{(x, q) Rd 1 : Pd 2 j 1 xj q, x Rd int(P)} Here P {x Rd : aiT x bi , 1 i m} P is locally flat at any y iff aiT y bi but ajT y bj j 6 i for some i. Bienstock, Michalka Convex obj non-convex domain Columbia
{(x, q) Rd 1 : Pd 2 j 1 xj q, x Rd int(P)} Here P {x Rd : aiT x bi , 1 i m} P is locally flat at any y iff aiT y bi but ajT y bj j 6 i for some i. Lifted “first-order” inequality: q 2y T x ky k2 α (aiT x bi ) for α 0 appropriately chosen. Bienstock, Michalka Convex obj non-convex domain Columbia
{(x, q) Rd 1 : Pd 2 j 1 xj q, x Rd int(P)} Here P {x Rd : aiT x bi , 1 i m} P is locally flat at any y iff aiT y bi but ajT y bj j 6 i for some i. Lifted “first-order” inequality: q 2y T x ky k2 α (aiT x bi ) for α 0 appropriately chosen. Theorem: Let (x̂, q̂) Rd 1 with v̂ int(P). We can compute a lifted first-order inequality maximally violated by (x̂, q̂), by solving m linearly constrained convex quadratic programs on O(d) variables. Bienstock, Michalka Convex obj non-convex domain Columbia
When does a point x̂, d X x̂j2 j 1 violate a lifted first-order inequality q 2y T x ky k2 α (aiT x bi ) ? Bienstock, Michalka Convex obj non-convex domain Columbia
When does a point x̂, d X x̂j2 j 1 violate a lifted first-order inequality q 2y T x ky k2 α (aiT x bi ) ? It does, if and only if: d X x̂j2 2y T x̂ αaiT x̂ ky k2 αb j 1 Bienstock, Michalka Convex obj non-convex domain Columbia
When does a point x̂, d X x̂j2 j 1 violate a lifted first-order inequality q 2y T x ky k2 α (aiT x bi ) ? It does, if and only if: d X x̂j2 2y T x̂ αaiT x̂ ky k2 αb j 1 This describes the interior of a ball, Bienstock, Michalka Convex obj non-convex domain Columbia
When does a point x̂, d X x̂j2 j 1 violate a lifted first-order inequality q 2y T x ky k2 α (aiT x bi ) ? It does, if and only if: d X x̂j2 2y T x̂ αaiT x̂ ky k2 αb j 1 This describes the interior of a ball, which must be contained in int(P) Bienstock, Michalka Convex obj non-convex domain Columbia
Geometrical characterization int(P) cut off region y Bienstock, Michalka Convex obj non-convex domain Columbia
Geometrical characterization int(P) cut off region y Characterization: x Rd int(P) iff, for each ball B(µ, R) P, Bienstock, Michalka Convex obj non-convex domain Columbia
Geometrical characterization int(P) cut off region y Characterization: x Rd int(P) iff, for each ball B(µ, R) P, kx µk2 R 2 Bienstock, Michalka Convex obj non-convex domain Columbia
S {(x, q) Rd 1 : Pd 2 j 1 xj q, x Rd int(P)} P {x Rd : aiT x bi , 1 i m} Bienstock, Michalka Convex obj non-convex domain Columbia
S {(x, q) Rd 1 : Pd 2 j 1 xj q, x Rd int(P)} P {x Rd : aiT x bi , 1 i m} conv (S) conv (Q1 Q2 . . . Qm ) , where Qi Bienstock, Michalka Convex obj non-convex domain (x, q) Rd 1 : d X j 1 xj2 q, aiT x bi . Columbia
S {(x, q) Rd 1 : Pd 2 j 1 xj q, x Rd int(P)} P {x Rd : aiT x bi , 1 i m} conv (S) conv (Q1 Q2 . . . Qm ) , where Qi (x, q) Rd 1 : d X j 1 xj2 q, aiT x bi . Ceria and Soares (1999) min { q : (x, q) S} is an SOCP Bienstock, Michalka Convex obj non-convex domain Columbia
S {(x, q) Rd 1 : Pd 2 j 1 xj q, x Rd int(P)} P {x Rd : aiT x bi , 1 i m} conv (S) conv (Q1 Q2 . . . Qm ) , where Qi (x, q) Rd 1 : d X xj2 q, aiT x bi j 1 . Ceria and Soares (1999) min { q : (x, q) S} is an SOCP with O(md) variables and m conic constraints Bienstock, Michalka Convex obj non-convex domain Columbia
S {(x, q) Rd 1 : Pd 2 j 1 xj q, x Rd int(P)} P {x Rd : aiT x bi , 1 i m} conv (S) conv (Q1 Q2 . . . Qm ) , where Qi (x, q) Rd 1 : d X xj2 q, aiT x bi j 1 . Ceria and Soares (1999) min { q : (x, q) S} is an SOCP with O(md) variables and m conic constraints Separation: solve SOCP, use SOCP “Farkas Lemma”, get linear cut Bienstock, Michalka Convex obj non-convex domain Columbia
Second setting: separating across a quadratic set For A 0, polynomially separable linear inequality description for: {(x, q) Rd 1 : d X xj2 q, x T Ax 2c T x b 0} j 1 P {x Rd : x T Ax 2c T x b 0} . Bienstock, Michalka Convex obj non-convex domain Columbia
Second setting: separating across a quadratic set For A 0, polynomially separable linear inequality description for: {(x, q) Rd 1 : d X xj2 q, x T Ax 2c T x b 0} j 1 P {x Rd : x T Ax 2c T x b 0} . P Bienstock, Michalka Convex obj non-convex domain µ ρ Columbia
Second setting: separating across a quadratic set For A 0, polynomially separable linear inequality description for: {(x, q) Rd 1 : d X xj2 q, x T Ax 2c T x b 0} j 1 P {x Rd : x T Ax 2c T x b 0} . P µ ρ Separation problem: given(x̂, q̂) Rd 1 , Find a ball B(µ, ρ) P, so as to maximize Bienstock, Michalka Convex obj non-convex domain Columbia
Second setting: separating across a quadratic set For A 0, polynomially separable linear inequality description for: {(x, q) Rd 1 : d X xj2 q, x T Ax 2c T x b 0} j 1 P {x Rd : x T Ax 2c T x b 0} . P µ ρ Separation problem: given(x̂, q̂) Rd 1 , Find a ball B(µ, ρ) P, so as to maximize T ρ (q̂ 2µ x̂ µT µ) Bienstock, Michalka Convex obj non-convex domain Columbia
Second setting: separating across a quadratic set For A 0, polynomially separable linear inequality description for: {(x, q) Rd 1 : d X xj2 q, x T Ax 2c T x b 0} j 1 P {x Rd : x T Ax 2c T x b 0} . P µ ρ Separation problem: given(x̂, q̂) Rd 1 , Find a ball B(µ, ρ) P, so as to maximize T ρ (q̂ 2µ x̂ µT µ) ρ kx̂ µk2 q̂ kx̂k2 Bienstock, Michalka Convex obj non-convex domain Columbia
Separation problem: min µ,ρ Subject to: Bienstock, Michalka Convex obj non-convex domain kµk2 ρ 2x̄ T µ {x : kx µk2 ρ} P Columbia
Separation problem: min µ,ρ Subject to: kµk2 ρ 2x̄ T µ {x : kx µk2 ρ} P Theorem: Optimal choices for µ and ρ are given by: µ̂ θ̂b (I θ̂A)x̄ and ρ̂ kµ̂k2 2x̄ T µ̂ kx̄k2 θ̂(x̄ T Ax̄ 2b T x̄ c). Here, θ̂ 1 λmax A . Bienstock, Michalka Convex obj non-convex domain Columbia
Separating accross general quadratics . Π { (x, w , z) Rd R R : z x T Qx q T x, w x T Ax } (A 0, Q 0). Bienstock, Michalka Convex obj non-convex domain Columbia
Separating accross general quadratics . Π { (x, w , z) Rd R R : z x T Qx q T x, w x T Ax } (A 0, Q 0). Linear transformation Π is the set of points Rd R R s.t. z kxk2 q T x, w x T Λx (Λ 0). . Write P {(x, w ) Rd R : x T Λx w 0}, and for µ Rd , ν R, . M(µ, ν) {(x, w ) Rd R : λmax kx µk2 (ν w ) 0}. Then x Rd int(P) iff x Rd int(M(µ, ν)), for all µ, ν with M(µ, ν) P. Bienstock, Michalka Convex obj non-convex domain Columbia
Π { (x, w , z) Rd R R : z kxk2 q T x, w x T Λ x}, P {(x, w ) Rd R : x T Λx w 0}, M(µ, ν) {(x, w ) Rd R : λmax kx µk2 (ν w ) 0}. Bienstock, Michalka Convex obj non-convex domain Columbia
Π { (x, w , z) Rd R R : z kxk2 q T x, w x T Λ x}, P {(x, w ) Rd R : x T Λx w 0}, M(µ, ν) {(x, w ) Rd R : λmax kx µk2 (ν w ) 0}. x Rd int(P) iff x Rd int(M(µ, ν)), µ, ν with M(µ, ν) P. Bienstock, Michalka Convex obj non-convex domain Columbia
Π { (x, w , z) Rd R R : z kxk2 q T x, w x T Λ x}, P {(x, w ) Rd R : x T Λx w 0}, M(µ, ν) {(x, w ) Rd R : λmax kx µk2 (ν w ) 0}. x Rd int(P) iff x Rd int(M(µ, ν)), µ, ν with M(µ, ν) P. So, valid inequality for any µ, ν with M(µ, ν) P: λmax kµk2 λmax (2µ q)T x (ν w ) λmax z 0 Separation problem, given (x̄, w̄ ) int(P) w̄ ν λmax kµk2 2λmax x̄ T µ 2λmax q T x̄ subject to: Bienstock, Michalka Convex obj non-convex domain ν min{ λmax kx µk2 x T Λx } 0. x Columbia
Separation problem, given (x̄, w̄ ) int(P) w̄ ν λmax kµk2 2λmax x̄ T µ 2λmax q T x̄ subject to: ν min{ λmax kx µk2 x T Λx } 0. x Lemma: This problem can be explicitly solved in polynomial time. Bienstock, Michalka Convex obj non-convex domain Columbia
Separation problem, given (x̄, w̄ ) int(P) w̄ ν λmax kµk2 2λmax x̄ T µ 2λmax q T x̄ subject to: ν min{ λmax kx µk2 x T Λx } 0. x Lemma: This problem can be explicitly solved in polynomial time. But we started with . Π { (x, w , z) Rd R R : z x T Qx q T x, w x T Ax } and transformed it into: Π { (x, w , z) Rd R R : z kxk2 q T x, w x T Λ x}, Bienstock, Michalka Convex obj non-convex domain Columbia
Separation problem, given (x̄, w̄ ) int(P) w̄ ν λmax kµk2 2λmax x̄ T µ 2λmax q T x̄ subject to: ν min{ λmax kx µk2 x T Λx } 0. x Lemma: This problem can be explicitly solved in polynomial time. But we started with . Π { (x, w , z) Rd R R : z x T Qx q T x, w x T Ax } and transformed it into: Π { (x, w , z) Rd R R : z kxk2 q T x, w x T Λ x}, Theorem. Eigenspace not necessary for poly-time separation (only max eigenvalue of A). Bienstock, Michalka Convex obj non-convex domain Columbia
. Example: f (x) 2(x1 x2 x1 x3 x2 x3 ) over [0, 1]3 . McCormick relaxation gives zero lower bound on f (x̄), where x̄ (1/2, 1/2, 1/2)T . Bienstock, Michalka Convex obj non-convex domain Columbia
. Example: f (x) 2(x1 x2 x1 x3 x2 x3 ) over [0, 1]3 . McCormick relaxation gives zero lower bound on f (x̄), where x̄ (1/2, 1/2, 1/2)T . (Kilinc, Linderoth, Luedtke) Bienstock, Michalka Convex obj non-convex domain Columbia
. Example: f (x) 2(x1 x2 x1 x3 x2 x3 ) over [0, 1]3 . McCormick relaxation gives zero lower bound on f (x̄), where x̄ (1/2, 1/2, 1/2)T . (Kilinc, Linderoth, Luedtke) However, f (x) U(x) L(x), where . U(x) (x1 x2 )2 (x1 x3 )2 (x2 x3 )2 , . L(x) 2(x12 x22 x32 ). Bienstock, Michalka Convex obj non-convex domain Columbia
. Example: f (x) 2(x1 x2 x1 x3 x2 x3 ) over [0, 1]3 . McCormick relaxation gives zero lower bound on f (x̄), where x̄ (1/2, 1/2, 1/2)T . (Kilinc, Linderoth, Luedtke) However, f (x) U(x) L(x), where . U(x) (x1 x2 )2 (x1 x3 )2 (x2 x3 )2 , . L(x) 2(x12 x22 x32 ). Above techniques, plus (for example) one disjunction prove f (x̄) .43599, using linear inequalities. Bienstock, Michalka Convex obj non-convex domain Columbia
. Example: f (x) 2(x1 x2 x1 x3 x2 x3 ) over [0, 1]3 . McCormick relaxation gives zero lower bound on f (x̄), where x̄ (1/2, 1/2, 1/2)T . (Kilinc, Linderoth, Luedtke) However, f (x) U(x) L(x), where . U(x) (x1 x2 )2 (x1 x3 )2 (x2 x3 )2 , . L(x) 2(x12 x22 x32 ). Above techniques, plus (for example) one disjunction prove f (x̄) .43599, using linear inequalities. Probably not best possible. Bienstock, Michalka Convex obj non-convex domain Columbia
. Example: f (x) 2(x1 x2 x1 x3 x2 x3 ) over [0, 1]3 . McCormick relaxation gives zero lower bound on f (x̄), where x̄ (1/2, 1/2, 1/2)T . (Kilinc, Linderoth, Luedtke) However, f (x) U(x) L(x), where . U(x) (x1 x2 )2 (x1 x3 )2 (x2 x3 )2 , . L(x) 2(x12 x22 x32 ). Above techniques, plus (for example) one disjunction prove f (x̄) .43599, using linear inequalities. Probably not best possible. Vielen Dank! Bienstock, Michalka Convex obj non-convex domain Columbia
Convex obj non-convex domain. Introduction Generic Problem: min Q(x); s:t: x 2F; Q(x) convex, especially: convex quadratic F nonconvex Examples: F is a mixed-integer set F is constrained in a nasty way, e.g. x 1 3 sin(x 2) 2cos(x 3) 4 Bienstock, Michalka Columbia Convex obj non-convex domain.
Solution. We prove the rst part. The intersection of two convex sets is convex. There-fore if Sis a convex set, the intersection of Swith a line is convex. Conversely, suppose the intersection of Swith any line is convex. Take any two distinct points x1 and x2 2 S. The intersection of Swith the line through x1 and x2 is convex.
What is convex projective geometry? Motivation from hyperbolic geometry Nice Properties of Hyperbolic Space Convex: Intersection with projective lines is connected. Properly Convex: Convex and closure is contained in an affine patch ()Disjoint from some projective hyperplane. Strictly Convex: Properly convex and boundary contains
3 convex-convex n [DK85] convex-convex (n;lognp lognq) [DK90] INTERSECTION DETECTION OF CONVEX POLYGONS Perhaps the most easily understood example of how the structure of geometric objects can be exploited to yield an e cient intersection test is that of detecting the intersection of two convex polygons. There are a number of solutions to this .
Proof:Let us denote the set of all convex combinations of ppoints of Sby Cp(S). Then the set of all possible convex combinations of points of S is C(S) : [1 p 1Cp(S). If x2 C(S) then it is a convex com
The optimization problem (1.1) is convex if every function involved f 0;f 1;:::;f m, is convex. The examples presented in section (1.1.2) are all convex. Examples of non-convex problems include combinatorial optimization problems, where (some if not all) variables are constrained to be bo
Convex optimization – Boyd & Vandenberghe Nonlinear programming – Bertsekas Convex Analysis – Rockafellar Fundamentals of convex analysis – Urruty, Lemarechal Lectures on modern convex optimization – Nemirovski Optimization for Machine Learning – Sra, Nowozin, Wright Theory of Convex Optimization for Machine Learning – Bubeck .
Convex Optimization Theory Athena Scientific, 2009 by Dimitri P. Bertsekas Massachusetts Institute of Technology Supplementary Chapter 6 on Convex Optimization Algorithms This chapter aims to supplement the book Convex Optimization Theory, Athena Scientific, 2009 with material on convex optimization algorithms. The chapter will be .
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