A Makefile with Multiple Executables

Some suggestions (assuming you use GNU make, not something else)

First, run once make -p, you'll understand what builtin rules make is knowing. Look in particular for COMPILE.c and LINK.c

Then, I suggest

 CFLAGS= -g -Wall -I.

^{(because you really want -g for debugging, and -Wall to get most warnings)}

And you probably don't need

$(EXECUTABLE): $(OBJ)
    gcc -o $@ $^ $(CFLAGS)

However, I suggest adding before most other rules

.PHONY: all clean

all: $(EXECUTABLES)

Actually, I would code your Makefile (for GNU make!) as follow

# file Makefile
CC= gcc
RM= rm -vf
CFLAGS= -Wall -g
CPPFLAGS= -I.
SRCFILES= ex1.c ex2.c ## or perhaps $(wildcard *.c)
OBJFILES= $(patsubst %.c, %.o, $(SRCFILES))
PROGFILES= $(patsubst %.c, %, $(SRCFILES))

.PHONY: all clean

all: $(PROGFILES)
clean:
     $(RM) $(OBJFILES) $(PROGFILES) *~
## eof Makefile

Remember that tab is a significant character in Makefile-s (action part of rules). In this answer, lines starting with four spaces at least should really start with a tab character.

Once everything is debugged consider running make clean to clean everything, and then make -j CFLAGS=-O2 all to compile in parallel everything with optimizations.

At last, I recommend using remake and running remake -x to debug complex Makefile-s

Of course, I'm supposing that your directory has only single-file programs.

BTW, there are other build automation tools. Perhaps you might consider using omake or ninja. For building large programs (millions of source code lines) consider also automake, ccache, cmake, icecream. In some cases, consider generating some C code with GPP, GNU bison, SWIG, etc... or using your own Python or Guile script (or C meta-program). See also this draft report.

^{Don't forget to use a version control system like git for your source files. It is also time to learn such a tool.}

The following answer includes multiple executable such as initiate, process1, process2, ..., process4.

LOCAL_INCLUDE=./

all: clean process_first process_second init

process_first:
    gcc -g -o process1  -I$(LOCAL_INCLUDE) process1.c  -lzmq  -L. -L./.
    gcc -g -o process2  -I$(LOCAL_INCLUDE) process2.c  -lzmq  -L. -L./.

process_second:
    gcc -g -o process3  -I$(LOCAL_INCLUDE) process3.c  -lzmq  -L. -L./.
    gcc -g -o process4  -I$(LOCAL_INCLUDE) process4.c  -lzmq  -L. -L./.

init:
    gcc -g -o initiate -I$(LOCAL_INCLUDE) initiate.c -lzmq -lconfig -lpthread -L. -L./. -ldl -lrt

clean:
    rm -rf init_manager.o init_manager
    rm -rf process1 process2 process3 process4

NOTE: It is a good practice to clean and touch all the executable files before making them again.

You're close, but you need a pattern rule:

$(EXECUTABLE): % : %.c

And then a default rule to make it build both:

all: $(EXECUTABLE)