Zero-Convex Functions, Perturbation Resilience, and Subgradient - - PowerPoint PPT Presentation
Zero-Convex Functions, Perturbation Resilience, and Subgradient Projections for Feasibility-Seeking Methods Daniel Reem (joint work with Yair Censor ) Department of Mathematics, The Technion, Haifa, Israel E-mail : dream@tx.technion.ac.il
Daniel Reem (joint work with Yair Censor)
Department of Mathematics, The Technion, Haifa, Israel E-mail: dream@tx.technion.ac.il http://w3.impa.br/~dream
1 July 2016: 14th EUROPT Workshop on Advances in Continuous Optimization, Warsaw, Poland
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 1 / 24
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 2 / 24
Given a family (Cj)j∈J of closed and convex subsets in a given space, say Rd, to compute (approximately) a point y ∈ C :=
Cj assuming C = ∅.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 2 / 24
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 3 / 24
Represents the solution set of a system of (convex) inequalities g1(x) ≤ . . . gn(x) ≤ whenever Cj = {x : gj(x) ≤ 0} for some function gj.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 3 / 24
Represents the solution set of a system of (convex) inequalities g1(x) ≤ . . . gn(x) ≤ whenever Cj = {x : gj(x) ≤ 0} for some function gj. Has been used in the analysis of various phenomena, including:
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 3 / 24
Represents the solution set of a system of (convex) inequalities g1(x) ≤ . . . gn(x) ≤ whenever Cj = {x : gj(x) ≤ 0} for some function gj. Has been used in the analysis of various phenomena, including:
sensor networks;
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 3 / 24
Represents the solution set of a system of (convex) inequalities g1(x) ≤ . . . gn(x) ≤ whenever Cj = {x : gj(x) ≤ 0} for some function gj. Has been used in the analysis of various phenomena, including:
sensor networks; computerized tomography;
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 3 / 24
Represents the solution set of a system of (convex) inequalities g1(x) ≤ . . . gn(x) ≤ whenever Cj = {x : gj(x) ≤ 0} for some function gj. Has been used in the analysis of various phenomena, including:
sensor networks; computerized tomography; data compression;
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 3 / 24
Represents the solution set of a system of (convex) inequalities g1(x) ≤ . . . gn(x) ≤ whenever Cj = {x : gj(x) ≤ 0} for some function gj. Has been used in the analysis of various phenomena, including:
sensor networks; computerized tomography; data compression; molecular biology (example in the paper);
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 3 / 24
Represents the solution set of a system of (convex) inequalities g1(x) ≤ . . . gn(x) ≤ whenever Cj = {x : gj(x) ≤ 0} for some function gj. Has been used in the analysis of various phenomena, including:
sensor networks; computerized tomography; data compression; molecular biology (example in the paper); many more
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 3 / 24
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 4 / 24
Mainly iterative algorithms (e.g., projections on the subsets).
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 4 / 24
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 5 / 24
an operation of the form
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 5 / 24
an operation of the form Aj(x) = x − αt, α ≥ 0, t ∈ ∂gj(x).
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 5 / 24
an operation of the form Aj(x) = x − αt, α ≥ 0, t ∈ ∂gj(x). less computational demanding than standard projection on a set.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 5 / 24
an operation of the form Aj(x) = x − αt, α ≥ 0, t ∈ ∂gj(x). less computational demanding than standard projection on a set.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 5 / 24
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 6 / 24
Meaning: convergence is conserved despite perturbations.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 6 / 24
Meaning: convergence is conserved despite perturbations. Motivation:
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 6 / 24
Meaning: convergence is conserved despite perturbations. Motivation: imprecision is inherent:
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 6 / 24
Meaning: convergence is conserved despite perturbations. Motivation: imprecision is inherent: computational errors, noise, etc.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 6 / 24
Meaning: convergence is conserved despite perturbations. Motivation: imprecision is inherent: computational errors, noise, etc. lack of proof:
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 6 / 24
Meaning: convergence is conserved despite perturbations. Motivation: imprecision is inherent: computational errors, noise, etc. lack of proof: resilience of many algorithms has not been proved.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 6 / 24
Meaning: convergence is conserved despite perturbations. Motivation: imprecision is inherent: computational errors, noise, etc. lack of proof: resilience of many algorithms has not been proved. Superiorization:
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 6 / 24
Meaning: convergence is conserved despite perturbations. Motivation: imprecision is inherent: computational errors, noise, etc. lack of proof: resilience of many algorithms has not been proved. Superiorization: a recent optimization methodology which uses perturbations in an active way in order to obtain “superior” solutions.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 6 / 24
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 7 / 24
Introducing and discussing in a quite detailed way the class of zero-convex functions, a rich class of functions holding a promising potential
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 7 / 24
Introducing and discussing in a quite detailed way the class of zero-convex functions, a rich class of functions holding a promising potential Discussing the SSP method for solving the CFP in a general setting:
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 7 / 24
Introducing and discussing in a quite detailed way the class of zero-convex functions, a rich class of functions holding a promising potential Discussing the SSP method for solving the CFP in a general setting:
zero-convex functions
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 7 / 24
Introducing and discussing in a quite detailed way the class of zero-convex functions, a rich class of functions holding a promising potential Discussing the SSP method for solving the CFP in a general setting:
zero-convex functions domain: closed and convex subset of a real Hilbert space
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 7 / 24
Introducing and discussing in a quite detailed way the class of zero-convex functions, a rich class of functions holding a promising potential Discussing the SSP method for solving the CFP in a general setting:
zero-convex functions domain: closed and convex subset of a real Hilbert space Certain perturbations are allowed without losing convergence
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 7 / 24
Introducing and discussing in a quite detailed way the class of zero-convex functions, a rich class of functions holding a promising potential Discussing the SSP method for solving the CFP in a general setting:
zero-convex functions domain: closed and convex subset of a real Hilbert space Certain perturbations are allowed without losing convergence infinitely many sets are allowed
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 7 / 24
Introducing and discussing in a quite detailed way the class of zero-convex functions, a rich class of functions holding a promising potential Discussing the SSP method for solving the CFP in a general setting:
zero-convex functions domain: closed and convex subset of a real Hilbert space Certain perturbations are allowed without losing convergence infinitely many sets are allowed general control sequence (beyond cyclic and almost cyclic)
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 7 / 24
Introducing and discussing in a quite detailed way the class of zero-convex functions, a rich class of functions holding a promising potential Discussing the SSP method for solving the CFP in a general setting:
zero-convex functions domain: closed and convex subset of a real Hilbert space Certain perturbations are allowed without losing convergence infinitely many sets are allowed general control sequence (beyond cyclic and almost cyclic)
Convergence:
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 7 / 24
Introducing and discussing in a quite detailed way the class of zero-convex functions, a rich class of functions holding a promising potential Discussing the SSP method for solving the CFP in a general setting:
zero-convex functions domain: closed and convex subset of a real Hilbert space Certain perturbations are allowed without losing convergence infinitely many sets are allowed general control sequence (beyond cyclic and almost cyclic)
Convergence: global and weak, sometimes also strong
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 7 / 24
Introducing and discussing in a quite detailed way the class of zero-convex functions, a rich class of functions holding a promising potential Discussing the SSP method for solving the CFP in a general setting:
zero-convex functions domain: closed and convex subset of a real Hilbert space Certain perturbations are allowed without losing convergence infinitely many sets are allowed general control sequence (beyond cyclic and almost cyclic)
Convergence: global and weak, sometimes also strong Computational simulations: for a problem in molecular biology
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 7 / 24
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 8 / 24
Definition
H is a real Hilbert space.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 8 / 24
Definition
H is a real Hilbert space. Ω ⊆ H is nonempty and convex.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 8 / 24
Definition
H is a real Hilbert space. Ω ⊆ H is nonempty and convex. Given g : Ω → R, its 0-level-set is g≤0 = {x ∈ Ω | g(x) ≤ 0}.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 8 / 24
Definition
H is a real Hilbert space. Ω ⊆ H is nonempty and convex. Given g : Ω → R, its 0-level-set is g≤0 = {x ∈ Ω | g(x) ≤ 0}. g is said to be zero-convex at the point y ∈ Ω if there exists a vector t ∈ H (called a 0-subgradient of g at y) satisfying g(y) + t, x − y ≤ 0 ∀x ∈ g≤0.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 8 / 24
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 9 / 24
The set of all 0-subgradients of g at y is called the zero-subdifferential of g at y and denoted by ∂0g(y).
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 9 / 24
The set of all 0-subgradients of g at y is called the zero-subdifferential of g at y and denoted by ∂0g(y). A function g satisfying g(y) + t, x − y ≤ 0 ∀x ∈ g≤0. for all y ∈ Ω will be called 0-convex.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 9 / 24
The set of all 0-subgradients of g at y is called the zero-subdifferential of g at y and denoted by ∂0g(y). A function g satisfying g(y) + t, x − y ≤ 0 ∀x ∈ g≤0. for all y ∈ Ω will be called 0-convex. Other notions of subdifferentials exist in the literature, e.g.,
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 9 / 24
The set of all 0-subgradients of g at y is called the zero-subdifferential of g at y and denoted by ∂0g(y). A function g satisfying g(y) + t, x − y ≤ 0 ∀x ∈ g≤0. for all y ∈ Ω will be called 0-convex. Other notions of subdifferentials exist in the literature, e.g.,
the standard subdifferential
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 9 / 24
The set of all 0-subgradients of g at y is called the zero-subdifferential of g at y and denoted by ∂0g(y). A function g satisfying g(y) + t, x − y ≤ 0 ∀x ∈ g≤0. for all y ∈ Ω will be called 0-convex. Other notions of subdifferentials exist in the literature, e.g.,
the standard subdifferential the Clarke subdifferential
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 9 / 24
The set of all 0-subgradients of g at y is called the zero-subdifferential of g at y and denoted by ∂0g(y). A function g satisfying g(y) + t, x − y ≤ 0 ∀x ∈ g≤0. for all y ∈ Ω will be called 0-convex. Other notions of subdifferentials exist in the literature, e.g.,
the standard subdifferential the Clarke subdifferential the Quasi-subdifferential
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 9 / 24
The set of all 0-subgradients of g at y is called the zero-subdifferential of g at y and denoted by ∂0g(y). A function g satisfying g(y) + t, x − y ≤ 0 ∀x ∈ g≤0. for all y ∈ Ω will be called 0-convex. Other notions of subdifferentials exist in the literature, e.g.,
the standard subdifferential the Clarke subdifferential the Quasi-subdifferential Mordukhovich’s Subdifferential
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 9 / 24
The set of all 0-subgradients of g at y is called the zero-subdifferential of g at y and denoted by ∂0g(y). A function g satisfying g(y) + t, x − y ≤ 0 ∀x ∈ g≤0. for all y ∈ Ω will be called 0-convex. Other notions of subdifferentials exist in the literature, e.g.,
the standard subdifferential the Clarke subdifferential the Quasi-subdifferential Mordukhovich’s Subdifferential etc.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 9 / 24
The set of all 0-subgradients of g at y is called the zero-subdifferential of g at y and denoted by ∂0g(y). A function g satisfying g(y) + t, x − y ≤ 0 ∀x ∈ g≤0. for all y ∈ Ω will be called 0-convex. Other notions of subdifferentials exist in the literature, e.g.,
the standard subdifferential the Clarke subdifferential the Quasi-subdifferential Mordukhovich’s Subdifferential etc.
Our one seems to be new.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 9 / 24
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 10 / 24
The hyperplane M = {x ∈ H : t, x − y = −g(y)} separates g≤0 and y:
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 10 / 24
The hyperplane M = {x ∈ H : t, x − y = −g(y)} separates g≤0 and y:
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 10 / 24
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 11 / 24
Proposition
If g is zero-convex, then its zero-level-set g≤0 is convex.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 11 / 24
Proposition
If g is zero-convex, then its zero-level-set g≤0 is convex. If g≤0 is closed and convex, then g is zero-convex. In fact, we have a formula for the 0-subgradients using separating hyperplanes.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 11 / 24
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 12 / 24
Example
Any convex function g : Rn → R
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 12 / 24
Example
Any convex function g : Rn → R
Example
Any nonpositive function g is 0-convex at each y with t = 0.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 12 / 24
Example
Any convex function g : Rn → R
Example
Any nonpositive function g is 0-convex at each y with t = 0.
Example
Any lower semiconrinuous quasiconvex function is zero-convex.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 12 / 24
Example
Any convex function g : Rn → R
Example
Any nonpositive function g is 0-convex at each y with t = 0.
Example
Any lower semiconrinuous quasiconvex function is zero-convex. Such functions frequently appear in generalized convexity theory.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 12 / 24
Example
Any convex function g : Rn → R
Example
Any nonpositive function g is 0-convex at each y with t = 0.
Example
Any lower semiconrinuous quasiconvex function is zero-convex. Such functions frequently appear in generalized convexity theory. In particular, certain quadratic functions in subsets of Rm (economics)
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 12 / 24
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 13 / 24
Example
Multivariate polynomials:
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 13 / 24
Example
Multivariate polynomials: e.g., g : R2 → R defined by g(x1, x2) = x2
1 + x2 2 − x4 1x4 2 + x6 1x6 2/4 − 0.3.
This g is zero-convex but not quasiconvex.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 13 / 24
Example
Multivariate polynomials: e.g., g : R2 → R defined by g(x1, x2) = x2
1 + x2 2 − x4 1x4 2 + x6 1x6 2/4 − 0.3.
This g is zero-convex but not quasiconvex.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 13 / 24
Example
The Voronoi function:
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 14 / 24
Example
The Voronoi function: p ∈ Ω and A ⊆ H are given.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 14 / 24
Example
The Voronoi function: p ∈ Ω and A ⊆ H are given. the distance d(p, A) between p and A is positive.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 14 / 24
Example
The Voronoi function: p ∈ Ω and A ⊆ H are given. the distance d(p, A) between p and A is positive. g : Ω → R is defined by g(x) := d(x, p) − d(x, A) ∀x ∈ Ω.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 14 / 24
Example
The Voronoi function: p ∈ Ω and A ⊆ H are given. the distance d(p, A) between p and A is positive. g : Ω → R is defined by g(x) := d(x, p) − d(x, A) ∀x ∈ Ω. g is zero-convex but usually not quasiconvex
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 14 / 24
Example
The Voronoi function: p ∈ Ω and A ⊆ H are given. the distance d(p, A) between p and A is positive. g : Ω → R is defined by g(x) := d(x, p) − d(x, A) ∀x ∈ Ω. g is zero-convex but usually not quasiconvex g≤0 is the Voronoi cell of p with respect to A.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 14 / 24
Example
The Voronoi function: p ∈ Ω and A ⊆ H are given. the distance d(p, A) between p and A is positive. g : Ω → R is defined by g(x) := d(x, p) − d(x, A) ∀x ∈ Ω. g is zero-convex but usually not quasiconvex g≤0 is the Voronoi cell of p with respect to A. Remark: Voronoi diagrams appear in numerous places in science and technology and have diverse applications.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 14 / 24
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 15 / 24
Algorithm
The Sequential Subgradient Projections (SSP) Method with Perturbations
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 15 / 24
Algorithm
The Sequential Subgradient Projections (SSP) Method with Perturbations Initialization: x0 ∈ Ω is arbitrary.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 15 / 24
Algorithm
The Sequential Subgradient Projections (SSP) Method with Perturbations Initialization: x0 ∈ Ω is arbitrary. Iterative Step: xn+1 = PΩ
gi(n)(xn) tn 2 tn + bn
if gi(n)(xn) > 0, xn, if gi(n)(xn) ≤ 0,
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 15 / 24
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 16 / 24
λn = relaxation parameters ∈ (ǫ1, 2 − ǫ2),
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 16 / 24
λn = relaxation parameters ∈ (ǫ1, 2 − ǫ2), tn = 0-subgradients ∈ ∂0gi(n)(xn)
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 16 / 24
λn = relaxation parameters ∈ (ǫ1, 2 − ǫ2), tn = 0-subgradients ∈ ∂0gi(n)(xn), bn = error terms.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 16 / 24
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 17 / 24
Mn=an arbitrary separating (closed) hyperplane between xn and g≤0
i(n)
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 17 / 24
Mn=an arbitrary separating (closed) hyperplane between xn and g≤0
i(n),
mn=the projection of xn on Mn. Then: xn+1 = (1 − λn)xn + λnmn + bn.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 17 / 24
Mn=an arbitrary separating (closed) hyperplane between xn and g≤0
i(n),
mn=the projection of xn on Mn. Then: xn+1 = (1 − λn)xn + λnmn + bn.
Figure: Illustration when 0 < λn < 1 and Ω = H.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 17 / 24
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 18 / 24
Control Sequence: more general than cyclic and almost cyclic
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 18 / 24
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 19 / 24
Condition
C =
Cj =
j
= ∅.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 19 / 24
Condition
C =
Cj =
j
= ∅.
Condition
Each function gj is 0-convex, uniformly continuous on closed and bounded subsets, and weakly sequential lower semicontinuous.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 19 / 24
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 20 / 24
Condition
For a fixed M > d(x0, C), the following inequality is satisfied bn ≤ min
ǫ1ǫ2h2
n
2(5M + 4hn)
∀n ∈ N,
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 20 / 24
Condition
For a fixed M > d(x0, C), the following inequality is satisfied bn ≤ min
ǫ1ǫ2h2
n
2(5M + 4hn)
∀n ∈ N, where hn = gi(n)(xn)/tn, if gi(n)(xn) > 0, 0, if gi(n)(xn) ≤ 0.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 20 / 24
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 21 / 24
Condition
There exists a K > 0 such that tn ≤ K for all n ∈ N.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 21 / 24
Condition
There exists a K > 0 such that tn ≤ K for all n ∈ N. Holds in many cases (examples mentioned in the paper).
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 21 / 24
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 22 / 24
Theorem
Under the above conditions, the algorithm converges weakly to a point y ∈ F := B[x0, 2M] ∩ C from any initial point x0.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 22 / 24
Theorem
Under the above conditions, the algorithm converges weakly to a point y ∈ F := B[x0, 2M] ∩ C from any initial point x0. If int(F) = ∅, then the convergence is strong.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 22 / 24
Theorem
Under the above conditions, the algorithm converges weakly to a point y ∈ F := B[x0, 2M] ∩ C from any initial point x0. If int(F) = ∅, then the convergence is strong. Clarification:
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 22 / 24
Theorem
Under the above conditions, the algorithm converges weakly to a point y ∈ F := B[x0, 2M] ∩ C from any initial point x0. If int(F) = ∅, then the convergence is strong. Clarification: B[x0, 2M] is the closed ball of radius 2M and center x0.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 22 / 24
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 23 / 24
Assume:
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 23 / 24
Assume: f : Ω → R is quasiconvex, uniformly continuous on bounded sets, etc;
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 23 / 24
Assume: f : Ω → R is quasiconvex, uniformly continuous on bounded sets, etc; C =
j∈J g≤0 j
;
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 23 / 24
Assume: f : Ω → R is quasiconvex, uniformly continuous on bounded sets, etc; C =
j∈J g≤0 j
; Goal: to find an α-approximate minimizer of f over C ⊆ Ω assuming α is an upper bound for inf f ;
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 23 / 24
Assume: f : Ω → R is quasiconvex, uniformly continuous on bounded sets, etc; C =
j∈J g≤0 j
; Goal: to find an α-approximate minimizer of f over C ⊆ Ω assuming α is an upper bound for inf f ; Solution: to apply the algorithm with g−1 = f − α (still quasiconvex and hence 0-convex) and gj, j ∈ J (now J ∪ {−1} is the new index set).
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 23 / 24
Assume: f : Ω → R is quasiconvex, uniformly continuous on bounded sets, etc; C =
j∈J g≤0 j
; Goal: to find an α-approximate minimizer of f over C ⊆ Ω assuming α is an upper bound for inf f ; Solution: to apply the algorithm with g−1 = f − α (still quasiconvex and hence 0-convex) and gj, j ∈ J (now J ∪ {−1} is the new index set). We obtain x ∈ C s.t. f (x) ≤ α.
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 23 / 24
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 24 / 24
The paper and the talk can be found online:
arXiv:1405.1501
http://w3.impa.br/~dream/talks
Censor, Reem (Haifa, Technion) 0-convex, perturbation, subgrad. proj. July 2016 24 / 24