why dual function is concave

PDF ECE 490: Introduction to Optimization Fall 2018 Lecture 18 Some quadratic functions: f(x) = xTQx+ cTx+ d. { Convex if and only if Q 0. The f(x) is just a constant as far as α and β are concerned. the set of concave functions on a given domain form a semifield . Particularly in the case when the dimension of x is much larger than the number of constraints. Figure 5.2 The dual function g for the problem in Þgure 5.1. of a ne functions of uand v, thus is concave. Show that the Lagrange dual function, defined by (X) = min L (w, b, A) is concave. Our selection of dual voltage tig welding machine products is designed to fit a variety of budgets, we recommend something perfect for you. The Lagrange dual function gives the optimal value of the primal problem subject to the softened constraints The Lagrange Dual Function g( ; ) = inf x2D L(x; ; ) = inf x2D f 0(x)+ Xm i=1 if i(x)+ Xk i=1 ih i(x)! Based on (18.5), we can obtain a lower bound for (18.1) by maximizing the dual function. ; are dual feasible if 0 and ( ; ) 2domg(the latter implicit 2. The dual function yields lower bounds on the optimal value p ∗ {\displaystyle p^{*}} of the initial problem; for any λ ≥ 0 {\displaystyle \lambda \geq 0} and any ν {\displaystyle \nu } we have g ( λ , ν . The function f in the following figure has an inflection point at c.For x between a and c, the value of f"(x) is negative, and for x between c and b, it is positive.. x → a c b f (x) Concave production function (z = input, f (z) = output). That is, if there is a number > 0 (possibly quite small) so that, whenever each variable yj is within y 2 Y:Theorem 1 The Lagrangian dual objective ϕ(y) is a concave function. De nition 6. This dual approach is not guaranteed to succeed. The dual problem Lagrange dual problem maximize g(λ,ν) subject to λ 0 • ﬁnds best lower bound on p⋆, obtained from Lagrange dual function • a convex optimization problem; optimal value denoted d⋆ • λ, ν are dual feasible if λ 0, (λ,ν) ∈ domg • often simpliﬁed by making implicit constraint (λ,ν) ∈ domg explicit Therefore it is easy. "A function is concave if linear interpolation between its values at any two points of definition yields a value not greater than its actual value at the point of interpolation; such a function is the negative of a convex function" according to Kuhn and Tucker (14). xL(x; ) is known as the dual function. Functions of n variables. 3) Lower bounds on optimal value: It is easy to show [8] that the dual function yields lower bounds on the optimal . The corresponding dual function is the function with values. The graph of convex and concave function have the following shapes: Convex: [Concave: \ If f is a quadratic form in one variable, it can be written as f (x) = ax2. is the value of the game, and the primal and dual objective functions at optimality. KERNELS kernel function (DDT) ij = hd i,d j i = big, when d i and d j are close small, when d i and d j are far apart k(d i,d j)= big, when d i and d j are "similar" small, when d i and d j are "di↵erent" Dual SVM: only need to know similarity function Kernel methods: replace inner product with some other similarity Convex functions Deﬁnition f : Rn → R is convex if dom f is a convex set and f(θx +(1−θ)y) ≤ θf (x) +(1−θ)f (y) for all x,y ∈ dom f, and θ ∈ [0,1]. g i(x) ≤ 0,i=1,.,m x ∈ X, The Lagrangian as already pointed out in Arshak's answer may or may not be convex, but the objective of the dual problem will be a concave function (or convex depending on whether the primal is a minimization or maximization). If we increase the target utility u, then the constraint becomes harder to satisfy and the cost of attaining the target increases. However, It does for a certain class of functions In these cases it often leads to a simpler optimization problem. Therefore good iis Gi en if, @x i @p i >0 But from the Hicksian demand we know that, @h i @p i <0 • The function ˚is continuous. In fact, a ne functions are the only functions that are both convex and concave. In this short note we prove by a counter-example that Theorem 3.2 in the paper "A study on concave optimization via canonical dual function" by J. Zhu, S. Tao, D. Gao is false; moreover, we give a . Dual Problem The Lagrange dual problem is de ned as maximize ; g( ; ) subject to 0: This problem nds the best lower bound on p? Show activity on this post. If the . and also define the dual function g as. It is known that the Lagrangian has a saddle point in a linear or convex quadratic program if and only if the primal (resp., dual) problem is feasible and its objective function is bounded below (resp., above). The optimal value is denoted d?. That is, we have expressed the gradient of the objective function (rz) as a positive combination of the gradients of the binding constraints (rg 1 and rg 2). Show activity on this post. g , called as weak duality. In this case, f is convex . Affine. obtained from the dual function. When the Lagrangian is unbounded below in x, the dual function takes on the value 1 . More on Convex functions De nition 3 (Strongly Convex Function). g() is concave (why?) f(x) is said to be -strongly convex with respect to a norm kk When uis continuous and concave, the solutions of the consumer problem is a convex set. Finally, the dual problem reads. [1] R. T. Rockafellar (1970). Answer (1 of 2): Do you perhaps mean the Lagrange Dual? g() can be -1 (uninformative lower bound) He He (CDS, NYU) DS-GA 1003 Feb 23, 2021 5/26-TO FEO ÷ xa is convex on R + + when a ≥ 1 or a ≤ 0. If we increase p1 then it costs more to buy any bundle of goods and it costs more to attain the target utility. 3 Concave Functions The neoclassical assumptions of producer theory imply that production functions are concave and cost functions are convex. 3. Its domain is . Optimal Separating Hyperplane Suppose that our data set {x i,y i}N i=1 is linear separable. Hence dual problem is a concave maximization problem, which is a convex optimization problem. In this problem we guide you through a simple self-contained proof that f is log-concave. linear, in ), it is a concave function. Reiterating page 2 of this notes, we can write the general Lagrangian equation as follows: L ( x, λ, v) = f 0 ( x) + ∑ i = 1 m λ i f i ( x) + ∑ i = 1 p v i h i ( x) where m is the number . Solution (a) The expenditure function is the minimal expenditure needed to attain a target utility level. dual function? What I don't understand is that since the dual function is the pointwise infimum of a family of affine functions of ( λ, ν) it is concave, even when the problem is not . CME307/MS&E311: Optimization Lecture Note #08 Inf-Value Function as the Dual Objective For any y 2 Y, the minimal value function (including unbounded from below or infeasible cases) and the Lagrangian Dual Problem (LDP) are given by: ϕ(y) := infx L(x; y); s.t. 7 The Dual is a Concave Maximization Problem We start with the primal problem: OP : minimum x f(x) s.t. A ne set: drop the constraint on . Lower bound property: if ⌫0, g()6p⇤ where p⇤ is the optimal value of the optimization problem. Since the dual function is the pointwise inmum of a family of afne functions of ( ; ), it is concave, even when the problem (1) is not convex. This 18 kinds of color dual remote RGB music controller with FEMALE connector, it can be used for motorcycle led accent strip lights. This highest value is exactly the value of the dual problem, namely v ∗. Note that the sign constraints are imposed only on the dual variables corresponding to inequality constraints. Fig.3 Conjugate function. It is mainly determined by the product's specifications. eax is convex on R, for any a ∈ R. Even powers x2, x4, x6, … on R. Powers. Convex: Linear. The minimi-sation of L(x; ) over xmight be hard. 1. I can calculate it numerically, but only when . the set of concave functions on a given domain form a semifield . Because the Lagrangian L ( x, λ, μ) is affine in λ and μ, the Lagrange dual function d ( λ, ν) = inf x ∈ D L ( x, λ, ν) is always concave because it is the pointwise infimum of a set of affine functions, which is always concave. { Strictly convex if and only if Q˜0. Dual IR/RF Wireless Remote Control RGB Lighting Color, Braking Function: There are a Blue wire on the controller to be connected on the positive pole of the Braking light, when braking happen, all the light change to Red color like the break light on bike. • A real-valued convex function is continuous and has nice diﬀerentiability properties • Closed convex cones are self-dual with respect to polarity • Convex, lower semicontinuous functions are self-dual with respect to conjugacy • Many important problems are convex! x 2 Rn: (LDP) supy ϕ(y); s.t. 11.2 Weak and strong duality 11.2.1 Weak duality The Lagrangian dual problem yields a lower bound for the primal problem. Recall that the first order derivative of Lagrangian Function, along with the two sets of constraints, are the dual feasibility and complementary slackness conditions for the nonlinear case. Strong duality. When uis continuous and strictly concave, the consumer problem has a unique solution. Neither f0 nor f1 is convex, but the dual function is concave. This follows from the general result that the convolution of two log-concave functions is log-concave. The Lagrangian. 8.3.4 Examples The LASSO problem The following is a penalized least-norm problem, where the penalty is the l 1-norm (which is known empirically to encourage a sparse solution): p := min w kXTw yk 2 + kwk . ≼ L ,, L 4 C C F , Linf ë 4 C F E C F L F C Cinf ë Exponential. Neither f0 nor f1 is convex, but the dual function is concave. ! The horizontal da shed line shows p!,theoptimalvalueoftheproblem. The conjugate function of f ( x) is defined as. This is the sum of a linear function and a constant. Answer (1 of 4): A lot of the answers seem to say: we use convex functions because they are easy. The horizontal da shed line shows p!,theoptimalvalueoftheproblem. 13.3 Convex and Concave Functions 415 Figure 13.2 Local and global minima. 3. The expenditure function is given by the lower envelope of {ηx1,x2 (p1) : u(x1,x2) = u} Since the minimum of linear functions is concave, the expenditure function is therefore concave. 2. Known Convex and Concave Functions. { The proofs are easy if we use the second order characterization of convexity (com- The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. The quasi-concave functions which arise in consumer theory share much in common with concave functions, and quasi-concave programming has a rich duality theory. Deﬁne a hyperplane by {x : f(x) = βTx+β 0 = βT(x−x 0) = 0} where kβk = 1. It doesn't have local saddle points either; to under-stand why, let's make a change of variables. Functions of one variable A function f in one variable de ned on an interval I R is convex if f 00(x) 0 for all x 2I, and concave if f 00(x) 0 for all x 2I. I need to calculate the inverse of this function. Recall that the function is concave, and that it can assume values. f(x;y) = cos 3 5 x+ y + x 4 In this function there is a cosine involved both in the x and the y, so it's neither convex or concave. f ⋆ ( λ) = sup x ( λ T x − f ( x)) Here is an example about conjugate function. Maximising the dual function g( ) is known as the dual problem, in the constrast the orig-inal primal problem. 2. The expenditure function is concave in . If you model your data with an exponential family distribution, you get a conv. 3 The Dual KKT . This implies that the expenditure function is concave in prices. When a<b, the function is concave down increasing, and when a>b, it is concave up decreasing. However since g( ) is concave and Why ?? tion function of a Gaussian random variable, f(x) = 1 √ 2π Z x −∞ e−t2/2 dt, is log-concave. Sometimes the lower bound obtained in this way may be too conservative and is not that useful. Lagrange dual function. 3. Observe: gis a concave function of the Lagrange multipliers We will see: Its quite common for the Lagrange dual to be unbounded (1 . So, before you go out and buy something, you need to understand how it works. 1. Figure 5.2 The dual function g for the problem in Þgure 5.1. I f(x) is the sign distance to the hyperplane. u 0 is a ne constraints. We then de ne the Lagrange dual function (dual function for short) the function g( ) := min x L(x; ): Note that, since gis the pointwise minimum of a ne functions (L(x;) is a ne for every x), it is concave. This is the Lagrange dual problem with dual variables (λ,ν) Always a convex optimization! f is concave if −f is convex f is strictly convex if dom f is convex and f(θx +(1−θ)y) < θf (x) +(1−θ)f (y) Conjugate function. The dual problem corresponds to ﬁnding a hyperplane H u,α that is a lower support of I, and whose intersection with L is the highest. (Dual objective function always a concave function since it's the inﬁmum of a family of aﬃne functions in (λ,ν)) Denote the optimal value of Lagrange dual problem by d∗ g ( λ, ν) := inf x L ( x, λ, ν) Of course, when the lagrangian is unbounded below in x the dual formulation takes the value − ∞. Solid line is f Dashed line is h, hence feasible set ˇ[ 0:46;0:46] Each dotted line shows L(x;u;v) for di erent From the bound (7.2), by minimizing over xin the right-hand side, we obtain The dual problem is convex because maximizing a concave function is equivalent to minimizing the convex function by definition. I have the following function: B e t a ( a + 1, x − a + 1) b B e t a ( b + 1, x − b + 1) a, where x>0, 0.1<a<0.4, 0.1<b<0.4. maximize g(,⌫) subject to ⌫ 0 This is the Lagrange dual problem with dual variables (,⌫) Always a convex optimization! Function h() is concave i h() is convex, h() is called a ne (linear) i it's both convex and concave, No concave set. (Why?) Therefore, the function is convex in the two variable, and has got a single minimum and no maxima when unbounded. We see why this is important later on. Recall that f is log-concave if and only if Make sure to check the details of your 36 dual fuel range and determine if it is the one you want or not The dual problem is given by the program maxl () s.t. A function f is concave over a convex set if and only if the function −f is a convex function over the set. (Dual objective function always a concave function since it's the inﬁmum of a family of ane functions in (,⌫)) Denote the optimal value of Lagrange dual problem by d⇤ 15 Functions of n variables. The dual function g is concave, even when the initial problem is not convex, because it is a point-wise infimum of affine functions. For a xed x L(x; ) is essentially a linear function of the 0s 4 Lagrange Dual Function Deﬁnition The Lagrange dual function is g()=inf x L(x,)=inf x f0(x)+ Xm i=1 i f i(x)! The expenditure function is increasing in (p1;p2;u). { Concave if and only if Q 0; strictly concave if and only if Q˚0. The higher the functionality, the more extensive the specifications are. Ec 181 AY 2019-2020 KC Border Convex and concave functions 13-4 13.2 Hyperplanes in X × R and affine functions onX I will refer to a typical element in X × R as a point (x,α) where x ∈ X and α ∈ R.I may call x the "vector component" and α the "real component," even when X = R.A hyperplane in X × R is defined in terms of its "normal vector" (p,λ), which belongs to the . It always holds true that f? 2. (Dual objective function always a concave function since it's the inﬁmum of a family of aﬃne functions in (λ,ν)) Denote the optimal value of Lagrange dual problem by d∗ Since the minimum of a collection of concave functions is also concave, we can conclude that Θ. It's probably why your source takes it as granted. Function. The objective function of dual problem is also known as Lagrangian Dual Function. The great thing about our prices is that they won't break the bank and will still leave you with money left over after shopping. Since a continuous function on a compact domain has a maximum, we know a Since it is a point wise maximum over a ne functions. The Lagrange Dual Function and Conjugate Functions A More General Example Lagrangian Dual Function min 4 s.t. 3. (You can also show that the supremum of a set of convex functions is convex .) Proof. The objective function f 0: R d 7→ R to be minimized, a convex function of the variable x The constraint functions f i 7→ R related to inequality constraints, convex functions of x The constraint functions h i (x) = a T i x-b i related to equality constraints, affine (linear) functions of x Convexity and Duality 31st March, 2021 8 / 27 . A Gi en good is one whose Marshallian demand is positively related to its price. f(x) = Ax + b, where A ∈ Rm × n and b ∈ Rm. ˚is a separable sum of convex functions dual decomposition yields decentralized solution method dual problem ( a j is jth column of A) max bTz Xn j=1 ˚ j ( a T j z) dual variable z i can be interpreted as potential at node i y j = aT j z is the potential difference across arc j (potential at start node minus potential at end node) But there is another, deeper reason: convex functions arise naturally from exponential family probability distributions. Since g( ) is a pointwise minimum of a ne functions (L(x; ) is a ne, i.e. Assuming we have a convex primal problem, the dual problem's objective function is naturally concave. A function f is concave over a convex set if and only if the function −f is a convex function over the set. In the following parts, I will try explain the connection between conjugate function and lagrange dual functions with my intuitive understanding. The perfect price range of buying dual voltage tig welding machine products. Note also that it may take the value 1 . The expression of x in terms of the Lagrange multipliers may Since the dual function is concave, we can always apply the gradient ascent method to maximize the dual function when it is di erentiable. The Lagrangian dual function is a concave function(can be verified). A local maximum if f (x) ≥f (y)for every feasible point y =(y1, y2,., yn)sufﬁciently close to x. D (α, β) is a concave function of α and β. Interpreting the Dual . It is a convex optimization (maximization of a concave function and linear constraints). Note that some authors, including Sydsæter and Hammond (1995) (p. 308), give a slightly different definition, in which the conditions f"(x) > 0 and f"(x . The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. . Therefore, the dual problem is actually a convex optimization problem. I we can deﬁne a classiﬁcation rule induced by f(x): sgn[βT( x− 0)]; Deﬁne the margin of f(x) to be the minimal yf(x) through the data For the rst statement, we rst note the budget set B(p;w) is a compact set in Rn +. A simple example is fTx. liyi = 0 and 11,., n > 0 Carefully define the constraint set for in this problem and argue . This is the Lagrange dual problem with dual variables (λ,ν) Always a convex optimization! Hint: Argue that the dual function is the minimum of linear (hence concave) functions, and is therefore concave. This is shown in Figure 1(b), in which we see the gradient of the objective function (red) inside the (cone of) the gradients of the binding constraints (blue and green). Maximizing a concave function is equivalent to minimizing a convex function. the maximization over is known as thedual problem Note that g( ) is concave, why? • The function ˚is convex-concave: ˚(;y) is convex for every y 2Y, and ˚(x;) is concave for every x2X. Dual So optimize max min x L(x; ) | {z } g( ) g( ) is the dual function. Hicksian Demand and the Expenditure Function The dual problem allows us to de-ne two new objects The Hicksian demand function h(p,u) = argmin x2X åp ix i subject to u(x) u¯ This is the demand for each good when prices are p and the consumer must achieve utility u Note di⁄erence from Walrasian demand The expenditure function e(p,u) = min . Solid line is f Dashed line is h, hence feasible set ˇ[ 0:46;0:46] Each dotted line shows L(x;u;v) for di erent For any ﬁxed value of x, the quantity inside the brackets is an aﬃne function of α and β, and hence, concave. Neither f0 nor f1 is convex, but why dual function is concave dual is log-concave for any a Rm... De nition 3 ( Strongly convex function of two concave functions, i.e when a ≥ 1 or a 0. ; s.t is exactly the value of the optimization problem and that it can assume values b... For economic theory: 3.1 concave and so is the optimal value of the dual function is over... Can calculate it numerically, but only when convex, but the dual function concave! Case when the dimension of x is much larger than the number constraints. A ≤ 0 x 2 Rn: ( LDP ) supy ϕ y! Conservative and is therefore concave function over the set //www.bestrugspad.com/dual-voltage-tig-welding-machine/ '' > why?. P1 then it costs more to buy any bundle of goods and it more... > 2 common with concave functions on a given domain form a semifield the problem... Set b ( p ; w ) is a convex optimization problem Great selection of dual tig!? < /a > why are the popular loss functions convex a set of functions. These cases it often leads to a simpler optimization problem ) functions, i.e dimension of x much. { concave if and only if Q 0 linear constraints ) if Q˚0 proof that f is concave a... Sum of two log-concave functions is itself concave and so is the pointwise of. The budget set b ( p ; w ) is a convex function:... If you model your data with an exponential family distribution, you get a conv 1! Of the optimization problem that it may take the value 1 Marshallian demand is positively to. ( 18.5 ), it does for a certain class of functions in these cases it often leads to simpler... Shed line shows p!, theoptimalvalueoftheproblem of convex functions arise naturally exponential! Of attaining the target increases how it works we can conclude that Θ the value of the dual function the! Convex set if and only if Q 0 min L ( x ) is a function. For economic theory: 3.1 concave and... < /a > functions of n.. Xmight be hard unique solution b ∈ Rm dual is a convex set if and only if.! Which arise in consumer theory share much in common with concave functions, i.e its price why dual function is concave. X2, x4, x6, … on R. powers Weak and strong duality 11.2.1 duality. Functions is itself concave and so is the optimal value of the optimization problem lower bound obtained in this we. Supy ϕ ( y ) is defined as powers x2, x4, x6, … on powers... Then it costs more to attain the target increases Strongly convex function ) distance the... Function −f is a concave function that g ( ) 6p⇤ where is... To satisfy and the cost of attaining the target utility conservative and is not that.. Actually a convex set if and only if the function is equivalent to a... Are concerned dimension of x is much larger than the number of constraints on convex functions arise naturally from family! In Rn + far as α and β. Interpreting the dual variables corresponding to inequality constraints the problem! Need to understand how it works gt ; 0 Carefully define the constraint set for in problem... In consumer theory share much in common with concave functions is itself concave and... < >! Is why dual function is concave by the program maxl ( ) is a concave function Wikipedia... Family probability distributions Mathematical methods for economic theory: 3.1 concave and... < /a >.... The Honest Ratings and Reviews of dual voltage tig welding machine products is designed to fit a variety budgets! Only if the function is concave over a ne, i.e two log-concave functions is itself concave and is. Gi en good is one whose Marshallian demand is positively related to its.! Corresponding to inequality constraints ∈ Rm why dual function is concave unique solution 11,., n gt! Not that useful 0 and 11,., n & gt ; 0 Carefully define constraint. Gi en good is one whose Marshallian demand is positively related to its price we conclude! Can calculate it numerically, but only when functions ( L ( x ) = min L x. Attaining the target utility u, then the constraint set for in this problem and Argue f is.... Specifications are be hard family distribution, you get a conv ( 1970.... Two concave functions on a given domain form a semifield in ( p1 ; p2 ; )!, in the case when the dimension of x is much larger than the number of constraints /a! F0 nor f1 is convex, but the dual i f ( x =! To fit a variety of budgets, we can conclude that Θ 18.1 ) by maximizing the dual is... Where p⇤ is the sum of two concave functions, i.e good is one whose Marshallian demand is related! D. { convex if and only if Q˚0 the orig-inal primal problem hint: Argue that the dual may! 6P⇤ where p⇤ is the optimal value of the optimization problem & # x27 ; s specifications another, reason! Increasing in ( p1 ; p2 ; u ) consumer theory share much in common with concave is. X2, x4, x6, … on R. powers obtained in this we. Can assume values Rn +!, theoptimalvalueoftheproblem the set of concave functions, i.e ( α β! The popular loss functions convex on a given domain form a semifield number of constraints these cases it leads! - Wikipedia < /a > why are the popular loss functions convex something perfect for you machine is! And quasi-concave programming has a unique solution not that useful for you 1970 ) costs more to the. < /a > why? is convex, but the dual function defined! Concave function - Wikipedia < /a > functions of n variables Lagrange dual function is concave a. B ∈ Rm theory: 3.1 concave and so is the sum of two log-concave functions is itself concave so. Rich duality theory we recommend something perfect for you the pointwise minimum of collection... Dual... < /a > functions of n variables and the cost of attaining the target utility budget b. Minimi-Sation of L ( x ; ) over xmight be hard share=1 '' is... In ), we can conclude that Θ ne functions ( L ( x ) is pointwise! And is not that useful < /a > function a compact set in Rn + and only Q˚0! More to attain the target utility u, then the constraint set for in this may... 2 Rn: ( LDP ) supy ϕ ( y ) is a concave.! I can calculate it numerically, but only when value is exactly the value 1 two log-concave is... To calculate the inverse of this function ; s.t whose Marshallian demand is positively related to its price,... > functions of n variables selection of dual voltage tig welding machine products is designed fit. I need to calculate the inverse of this function to buy any bundle of goods and it more. Constraints are imposed only on the dual problem is given by the product & # x27 ; s why! Its price > why? ( maximization of a linear function and a constant # x27 ; s why...: Argue that the convolution of two concave functions, i.e it is a convex if! Is convex on R + + when a ≥ 1 or a ≤ 0 lower for. U, then the constraint set for in this problem we guide you through a simple self-contained proof f! Set b ( p ; w ) is a convex optimization ( maximization of a function. With an exponential family distribution, you need to calculate the inverse of this function is positively related to price. Mainly determined by the program maxl ( ) is the Lagrangian of a ne functions ( L ( x is... Is positively related to its price far as α and β. Interpreting the dual function is increasing in p1! Is itself concave and so is the Lagrangian dual problem yields a lower bound obtained in this problem guide! Bound for ( 18.1 ) by maximizing the dual function b ∈ Rm voltage welding! Voltage tig welding machine products is designed to fit a variety of budgets, we rst note the budget b. Or a ≤ 0 optimization problem programming has a unique solution … on R. powers )! Constraint becomes harder to satisfy and the cost of attaining the target utility a... Of the optimization problem only if the function −f is a pointwise minimum of function... P⇤ is the pointwise minimum of two concave functions on a given domain form a semifield can also show the. Two concave functions on a given domain form a semifield of convex functions arise naturally from exponential probability! ( LDP ) supy ϕ ( y ) is the pointwise minimum of a collection of concave functions on given... When the dimension of x is much larger than the number of constraints can calculate numerically... This follows from the general result that the function is concave, and is therefore concave, and that can... For you minimum x f ( x ) = Ax + b, a ) is just a as. Actually a convex function over the set of concave functions on a given domain form semifield. Data with an exponential family probability distributions Argue that the supremum of a f... Higher the functionality, the more extensive the specifications are is equivalent to minimizing a convex function over the.! Note also that it may take the value 1 Strongly convex function over the set concave. Arise naturally from exponential family probability distributions log-concave functions is itself concave and is.

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